:: CHORD semantic presentation

REAL is non empty non trivial non finite complex-membered ext-real-membered real-membered V141() V148() V149() V151() set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V146() V148() Element of bool REAL
bool REAL is non empty non trivial non finite set
COMPLEX is non empty non trivial non finite complex-membered V141() set
omega is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V146() V148() set
bool omega is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
RAT is non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V141() set
INT is non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V141() set
{} is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() set
the empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() set is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
3 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
_GraphSelectors is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V148() Element of bool NAT
VertexSelector is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
EdgeSelector is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
SourceSelector is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
TargetSelector is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
4 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
{VertexSelector,EdgeSelector,SourceSelector,TargetSelector} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
Seg 1 is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
dom {} is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() set
rng {} is empty trivial Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() set
G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
G - 1 is complex V25() integer ext-real set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(0 + 1) - 1 is complex V25() integer ext-real set
G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
G - 1 is complex V25() integer even ext-real set
S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S is complex V25() integer ext-real set
G is complex V25() integer ext-real set
G + 2 is complex V25() integer ext-real set
P is complex V25() integer ext-real set
P - G is complex V25() integer ext-real set
(P - G) + 1 is complex V25() integer ext-real set
P - S is complex V25() integer ext-real set
(P - S) + 1 is complex V25() integer ext-real set
((P - S) + 1) + 2 is complex V25() integer ext-real set
((P - G) + 1) - 3 is complex V25() integer ext-real set
(P - S) + 3 is complex V25() integer ext-real set
((P - S) + 3) - 3 is complex V25() integer ext-real set
P - (G + 2) is complex V25() integer ext-real set
S is non empty complex V25() integer non even ext-real set
G is non empty complex V25() integer non even ext-real set
S - 2 is complex V25() integer ext-real set
G + 1 is complex V25() integer even ext-real set
- 1 is complex V25() integer ext-real non positive set
(G + 1) + (- 1) is complex V25() integer ext-real set
S + (- 1) is complex V25() integer ext-real set
S - 1 is complex V25() integer even ext-real set
(S - 1) + (- 1) is complex V25() integer ext-real set
G is non empty complex V25() integer non even ext-real set
S is non empty complex V25() integer non even ext-real set
S + 2 is non empty complex V25() integer non even ext-real set
S + 1 is complex V25() integer even ext-real set
(S + 1) + 1 is non empty complex V25() integer non even ext-real set
G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(1 + 2) - 2 is complex V25() integer ext-real set
G - 2 is complex V25() integer ext-real set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
G - (2 * 1) is non empty complex V25() integer non even ext-real set
S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
2 * 2 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
6 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
5 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
2 * 3 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
5 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
4 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
4 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
8 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
7 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
2 * 4 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
7 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
2 * 2 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
4 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
2 * 3 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 3) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
5 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
5 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
G + P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
2 * x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
G + (2 * x) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
G . S is set
(G . S) .. G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
rng G is finite set
G . ((G . S) .. G) is set
G is non empty Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng G is non empty finite set
bool (rng G) is non empty finite V32() set
S is non empty finite Element of bool (rng G)
P is Element of S
P .. G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x is set
x .. G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
S is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
len G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(S,(len G)) -cut G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((S,(len G)) -cut G) is finite set
G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng G is finite set
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
S is set
S .. G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
P is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
(G,P) is finite set
len G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(P,(len G)) -cut G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((P,(len G)) -cut G) is finite set
G . (S .. G) is set
dom ((P,(len G)) -cut G) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
((P,(len G)) -cut G) . x is set
P + x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
G . x is set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
P + x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
len ((P,(len G)) -cut G) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len ((P,(len G)) -cut G)) + P is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len G) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x + P is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
- P is complex V25() integer ext-real non positive set
(x + P) + (- P) is complex V25() integer ext-real set
(len G) + (- P) is complex V25() integer ext-real set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len G) - P is complex V25() integer ext-real set
((len G) - P) + 1 is complex V25() integer ext-real set
((P,(len G)) -cut G) . (x + 1) is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
P is set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
<*P*> ^ S is non empty Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(G,(x + 1)) is finite set
len G is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
((x + 1),(len G)) -cut G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng (((x + 1),(len G)) -cut G) is finite set
(S,x) is finite set
len S is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(x,(len S)) -cut S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((x,(len S)) -cut S) is finite set
len <*P*> is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + (len S) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len (((x + 1),(len G)) -cut G) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len (((x + 1),(len G)) -cut G)) + (x + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len G) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len ((x,(len S)) -cut S) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len ((x,(len S)) -cut S)) + x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
dom (((x + 1),(len G)) -cut G) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
Seg (len ((x,(len S)) -cut S)) is finite len ((x,(len S)) -cut S) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
dom ((x,(len S)) -cut S) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
(len <*P*>) + x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G . ((len <*P*>) + x) is set
S . x is set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G . (x + 1) is set
x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
(len G) - x is complex V25() integer ext-real set
0 + ((len G) - x) is complex V25() integer ext-real set
- 1 is complex V25() integer ext-real non positive set
(- 1) + x is complex V25() integer ext-real set
1 + (- 1) is complex V25() integer ext-real set
x + (- 1) is complex V25() integer ext-real set
x - 1 is complex V25() integer ext-real set
x + x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
x + ((len G) - x) is complex V25() integer ext-real set
n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
n + x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
(n + x) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((n + x) + 1) + (- 1) is complex V25() integer ext-real set
((len S) + 1) + (- 1) is complex V25() integer ext-real set
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x + n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
(((x + 1),(len G)) -cut G) . x is set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(((x + 1),(len G)) -cut G) . (n + 1) is set
(x + 1) + n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G . ((x + 1) + n) is set
(x + n) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G . ((x + n) + 1) is set
S . (x + n) is set
((x,(len S)) -cut S) . (n + 1) is set
((x,(len S)) -cut S) . x is set
x is set
n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
(((x + 1),(len G)) -cut G) . n is set
((x,(len S)) -cut S) . n is set
n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
((x,(len S)) -cut S) . n is set
(((x + 1),(len G)) -cut G) . n is set
G is set
S is Relation-like NAT -defined G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of G
bool S is non empty finite V32() set
len S is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
P is Relation-like NAT -defined G -valued Function-like finite FinSubsequence-like Element of bool S
Seq P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (Seq P) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
card P is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
S is Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
x is set
x is set
H is set
n is non empty Element of bool (the_Vertices_of G)
G .edgesBetween n is Element of bool (the_Edges_of G)
G .edgesInto n is Element of bool (the_Edges_of G)
G .edgesOutOf n is Element of bool (the_Edges_of G)
(G .edgesInto n) /\ (G .edgesOutOf n) is Element of bool (the_Edges_of G)
the_Edges_of P is Element of bool (the_Edges_of G)
P . EdgeSelector is set
the_Target_of P is Relation-like the_Edges_of P -defined the_Vertices_of P -valued Function-like V39( the_Edges_of P) V40( the_Edges_of P, the_Vertices_of P) Element of bool [:(the_Edges_of P),(the_Vertices_of P):]
the_Edges_of P is set
the_Vertices_of P is non empty set
P . VertexSelector is set
[:(the_Edges_of P),(the_Vertices_of P):] is Relation-like set
bool [:(the_Edges_of P),(the_Vertices_of P):] is non empty set
P . TargetSelector is set
the_Target_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
[:(the_Edges_of G),(the_Vertices_of G):] is Relation-like set
bool [:(the_Edges_of G),(the_Vertices_of G):] is non empty set
G . TargetSelector is set
(the_Target_of G) | (the_Edges_of P) is Relation-like the_Edges_of P -defined the_Edges_of G -defined the_Vertices_of G -valued Function-like set
(the_Target_of P) . H is set
(the_Target_of G) . H is set
the_Source_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
G . SourceSelector is set
(the_Source_of G) . H is set
the_Source_of P is Relation-like the_Edges_of P -defined the_Vertices_of P -valued Function-like V39( the_Edges_of P) V40( the_Edges_of P, the_Vertices_of P) Element of bool [:(the_Edges_of P),(the_Vertices_of P):]
P . SourceSelector is set
(the_Source_of G) | (the_Edges_of P) is Relation-like the_Edges_of P -defined the_Edges_of G -defined the_Vertices_of G -valued Function-like set
(the_Source_of P) . H is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(S .edgeSeq())) is finite Element of bool (the_Edges_of G)
card (S .edges()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
2 * (card (S .edges())) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (card (S .edges()))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len (S .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (len (S .edgeSeq())) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (len (S .edgeSeq()))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Element of bool (the_Vertices_of G)
(the_Vertices_of G) \ S is Element of bool (the_Vertices_of G)
G .edgesBetween ((the_Vertices_of G) \ S) is Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
G .edgesInto ((the_Vertices_of G) \ S) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ S) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ S)) /\ (G .edgesOutOf ((the_Vertices_of G) \ S)) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ S,G .edgesBetween ((the_Vertices_of G) \ S)
the_Vertices_of P is non empty set
P . VertexSelector is set
the_Edges_of P is set
P . EdgeSelector is set
(the_Vertices_of P) \/ (the_Edges_of P) is non empty set
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x .last() is Element of the_Vertices_of G
x . (len x) is set
the_Edges_of P is Element of bool (the_Edges_of G)
x .edges() is finite Element of bool (the_Edges_of G)
x .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(x .edgeSeq())) is finite Element of bool (the_Edges_of G)
x .vertices() is finite Element of bool (the_Vertices_of G)
x .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(x .vertexSeq())) is finite Element of bool (the_Vertices_of G)
G .edgesBetween (x .vertices()) is Element of bool (the_Edges_of G)
G .edgesInto (x .vertices()) is Element of bool (the_Edges_of G)
G .edgesOutOf (x .vertices()) is Element of bool (the_Edges_of G)
(G .edgesInto (x .vertices())) /\ (G .edgesOutOf (x .vertices())) is Element of bool (the_Edges_of G)
the_Vertices_of P is non empty Element of bool (the_Vertices_of G)
x is set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . n is set
G .edgesBetween (the_Vertices_of P) is Element of bool (the_Edges_of G)
G .edgesInto (the_Vertices_of P) is Element of bool (the_Edges_of G)
G .edgesOutOf (the_Vertices_of P) is Element of bool (the_Edges_of G)
(G .edgesInto (the_Vertices_of P)) /\ (G .edgesOutOf (the_Vertices_of P)) is Element of bool (the_Edges_of G)
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is set
P is set
{S,P} is non empty finite set
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
x .vertices() is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
x .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(x .vertexSeq())) is finite Element of bool (the_Vertices_of G)
x .first() is Element of the_Vertices_of G
x . 1 is set
x is set
{x} is non empty trivial finite 1 -element set
n is set
{S,P} \ {x} is finite Element of bool {S,P}
bool {S,P} is non empty finite V32() set
x .find n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . (x .find n) is set
y is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
y + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . y is set
y + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . (y + 1) is set
{P} is non empty trivial finite 1 -element set
{S} is non empty trivial finite 1 -element set
{S,P} \ {S} is finite Element of bool {S,P}
{S} is non empty trivial finite 1 -element set
{P} is non empty trivial finite 1 -element set
{S,P} \ {P} is finite Element of bool {S,P}
x is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
the_Vertices_of P is non empty set
P . VertexSelector is set
the_Edges_of P is set
P . EdgeSelector is set
(the_Vertices_of P) \/ (the_Edges_of P) is non empty set
the_Vertices_of P is non empty Element of bool (the_Vertices_of G)
the_Edges_of P is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
x .vertices() is finite Element of bool (the_Vertices_of G)
x .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(x .vertexSeq())) is finite Element of bool (the_Vertices_of G)
x .edges() is finite Element of bool (the_Edges_of G)
x .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(x .edgeSeq())) is finite Element of bool (the_Edges_of G)
G .edgesBetween (x .vertices()) is Element of bool (the_Edges_of G)
G .edgesInto (x .vertices()) is Element of bool (the_Edges_of G)
G .edgesOutOf (x .vertices()) is Element of bool (the_Edges_of G)
(G .edgesInto (x .vertices())) /\ (G .edgesOutOf (x .vertices())) is Element of bool (the_Edges_of G)
G .edgesBetween (the_Vertices_of P) is Element of bool (the_Edges_of G)
G .edgesInto (the_Vertices_of P) is Element of bool (the_Edges_of G)
G .edgesOutOf (the_Vertices_of P) is Element of bool (the_Edges_of G)
(G .edgesInto (the_Vertices_of P)) /\ (G .edgesOutOf (the_Vertices_of P)) is Element of bool (the_Edges_of G)
G is Relation-like NAT -defined Function-like finite [Graph-like] set
S is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
the_Vertices_of S is non empty set
S . VertexSelector is set
the_Edges_of S is set
S . EdgeSelector is set
(the_Vertices_of S) \/ (the_Edges_of S) is non empty set
P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
x is Relation-like NAT -defined (the_Vertices_of S) \/ (the_Edges_of S) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of S
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P .first() is Element of the_Vertices_of G
P . 1 is set
P .last() is Element of the_Vertices_of G
len P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (len P) is set
x .first() is Element of the_Vertices_of S
x . 1 is set
x .last() is Element of the_Vertices_of S
x . (len x) is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . P is set
S . x is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
S .vertices() is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
S .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(S .vertexSeq())) is finite Element of bool (the_Vertices_of G)
S .first() is Element of the_Vertices_of G
S . 1 is set
P is set
x is set
x is set
G .walkOf (P,x,x) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
(G .walkOf (P,x,x)) .append S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(G .walkOf (P,x,x)) .last() is Element of the_Vertices_of G
len (G .walkOf (P,x,x)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(G .walkOf (P,x,x)) . (len (G .walkOf (P,x,x))) is set
Seg 3 is non empty finite 3 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
Seg (len (G .walkOf (P,x,x))) is non empty finite len (G .walkOf (P,x,x)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
dom (G .walkOf (P,x,x)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
((G .walkOf (P,x,x)) .append S) . 3 is set
(G .walkOf (P,x,x)) . 3 is set
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
y is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
len ((G .walkOf (P,x,x)) .append S) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((G .walkOf (P,x,x)) .append S) . y is set
((G .walkOf (P,x,x)) .append S) . C is set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((G .walkOf (P,x,x)) .append S) .first() is Element of the_Vertices_of G
((G .walkOf (P,x,x)) .append S) . 1 is set
(G .walkOf (P,x,x)) .first() is Element of the_Vertices_of G
(G .walkOf (P,x,x)) . 1 is set
dom ((G .walkOf (P,x,x)) .append S) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len (G .walkOf (P,x,x))) + C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . b is set
Seg (len ((G .walkOf (P,x,x)) .append S)) is non empty finite len ((G .walkOf (P,x,x)) .append S) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
dom ((G .walkOf (P,x,x)) .append S) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len (G .walkOf (P,x,x))) + C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
C + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (C + 1) is set
v is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len (G .walkOf (P,x,x))) + v is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
n1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
bg is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
v + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (v + 1) is set
y is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((G .walkOf (P,x,x)) .append S) . y is set
((G .walkOf (P,x,x)) .append S) . C is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
S .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(S .edgeSeq())) is finite Element of bool (the_Edges_of G)
P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
P .edges() is finite Element of bool (the_Edges_of G)
P .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(P .edgeSeq())) is finite Element of bool (the_Edges_of G)
S .vertices() is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
S .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(S .vertexSeq())) is finite Element of bool (the_Vertices_of G)
P .vertices() is finite Element of bool (the_Vertices_of G)
P .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(P .vertexSeq())) is finite Element of bool (the_Vertices_of G)
(S .vertices()) /\ (P .vertices()) is finite Element of bool (the_Vertices_of G)
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (len S) is set
{(S .first()),(S .last())} is non empty finite Element of bool (the_Vertices_of G)
P .first() is Element of the_Vertices_of G
P . 1 is set
P .last() is Element of the_Vertices_of G
len P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (len P) is set
S .append P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .append P) .first() is Element of the_Vertices_of G
(S .append P) . 1 is set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
(S .append P) . x is set
S . x is set
len (S .append P) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
dom (S .append P) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
(len (S .append P)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- 1 is complex V25() integer ext-real non positive set
((len (S .append P)) + 1) + (- 1) is complex V25() integer ext-real set
(len S) + (len P) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) + (len P)) + (- 1) is complex V25() integer ext-real set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . x is set
(S .append P) . x is set
S . n is set
(S .append P) . n is set
(S .append P) . x is set
S . x is set
H is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len S) + H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
y + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
(S .append P) . n is set
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (H + 1) is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . ((2 * 0) + 1) is set
P . C is set
S . ((2 * 0) + 1) is set
(S .append P) . x is set
(S .append P) . n is set
H is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len S) + H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
y + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len S) + C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
(S .append P) . ((len S) + H) is set
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (H + 1) is set
(S .append P) . ((len S) + C) is set
C + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (C + 1) is set
(S .append P) . x is set
(S .append P) . n is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- (len S) is complex V25() integer ext-real non positive set
((len S) + (len P)) + (- (len S)) is complex V25() integer ext-real set
(len S) + (- (len S)) is complex V25() integer ext-real set
(S .append P) .last() is Element of the_Vertices_of G
(S .append P) . (len (S .append P)) is set
(S .edgeSeq()) ^ (P .edgeSeq()) is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the_Edges_of G
(S .append P) .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .last() is Element of the_Vertices_of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (len S) is set
P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
P .first() is Element of the_Vertices_of G
P . 1 is set
S .append P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .append P) .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(S .append P) .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len ((S .append P) .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (S .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
P .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
P .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (P .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(S .length()) + (P .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(S .edgeSeq()) ^ (P .edgeSeq()) is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the_Edges_of G
len ((S .edgeSeq()) ^ (P .edgeSeq())) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
P is non empty Element of bool (the_Vertices_of G)
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
G .edgesBetween P is Element of bool (the_Edges_of G)
G .edgesInto P is Element of bool (the_Edges_of G)
G .edgesOutOf P is Element of bool (the_Edges_of G)
(G .edgesInto P) /\ (G .edgesOutOf P) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
x .edgesBetween P is Element of bool (the_Edges_of x)
the_Edges_of x is set
x . EdgeSelector is set
bool (the_Edges_of x) is non empty set
x .edgesInto P is Element of bool (the_Edges_of x)
x .edgesOutOf P is Element of bool (the_Edges_of x)
(x .edgesInto P) /\ (x .edgesOutOf P) is Element of bool (the_Edges_of x)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of x,P,x .edgesBetween P
the_Vertices_of x is non empty Element of bool (the_Vertices_of G)
x . VertexSelector is set
the_Vertices_of x is non empty Element of bool (the_Vertices_of x)
the_Vertices_of x is non empty set
bool (the_Vertices_of x) is non empty set
x . VertexSelector is set
n is set
the_Target_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
[:(the_Edges_of G),(the_Vertices_of G):] is Relation-like set
bool [:(the_Edges_of G),(the_Vertices_of G):] is non empty set
G . TargetSelector is set
(the_Target_of G) . n is set
the_Source_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
G . SourceSelector is set
(the_Source_of G) . n is set
the_Edges_of x is Element of bool (the_Edges_of G)
the_Target_of x is Relation-like the_Edges_of x -defined the_Vertices_of x -valued Function-like V39( the_Edges_of x) V40( the_Edges_of x, the_Vertices_of x) Element of bool [:(the_Edges_of x),(the_Vertices_of x):]
[:(the_Edges_of x),(the_Vertices_of x):] is Relation-like set
bool [:(the_Edges_of x),(the_Vertices_of x):] is non empty set
x . TargetSelector is set
(the_Target_of x) . n is set
the_Source_of x is Relation-like the_Edges_of x -defined the_Vertices_of x -valued Function-like V39( the_Edges_of x) V40( the_Edges_of x, the_Vertices_of x) Element of bool [:(the_Edges_of x),(the_Vertices_of x):]
x . SourceSelector is set
(the_Source_of x) . n is set
the_Edges_of x is Element of bool (the_Edges_of x)
x . EdgeSelector is set
n is set
H is set
n is set
the_Target_of x is Relation-like the_Edges_of x -defined the_Vertices_of x -valued Function-like V39( the_Edges_of x) V40( the_Edges_of x, the_Vertices_of x) Element of bool [:(the_Edges_of x),(the_Vertices_of x):]
the_Edges_of x is set
the_Vertices_of x is non empty set
[:(the_Edges_of x),(the_Vertices_of x):] is Relation-like set
bool [:(the_Edges_of x),(the_Vertices_of x):] is non empty set
x . TargetSelector is set
(the_Target_of x) . n is set
(the_Target_of x) . n is set
the_Source_of x is Relation-like the_Edges_of x -defined the_Vertices_of x -valued Function-like V39( the_Edges_of x) V40( the_Edges_of x, the_Vertices_of x) Element of bool [:(the_Edges_of x),(the_Vertices_of x):]
x . SourceSelector is set
(the_Source_of x) . n is set
(the_Source_of x) . n is set
(the_Source_of G) . n is set
(the_Target_of G) . n is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
P is non empty Element of bool (the_Vertices_of G)
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
G .edgesBetween P is Element of bool (the_Edges_of G)
G .edgesInto P is Element of bool (the_Edges_of G)
G .edgesOutOf P is Element of bool (the_Edges_of G)
(G .edgesInto P) /\ (G .edgesOutOf P) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
x .edgesBetween P is Element of bool (the_Edges_of x)
the_Edges_of x is set
x . EdgeSelector is set
bool (the_Edges_of x) is non empty set
x .edgesInto P is Element of bool (the_Edges_of x)
x .edgesOutOf P is Element of bool (the_Edges_of x)
(x .edgesInto P) /\ (x .edgesOutOf P) is Element of bool (the_Edges_of x)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,P,G .edgesBetween P
the_Edges_of x is Element of bool (the_Edges_of G)
x . EdgeSelector is set
the_Edges_of x is Element of bool (the_Edges_of G)
n is set
the_Target_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
[:(the_Edges_of G),(the_Vertices_of G):] is Relation-like set
bool [:(the_Edges_of G),(the_Vertices_of G):] is non empty set
G . TargetSelector is set
(the_Target_of G) . n is set
the_Target_of x is Relation-like the_Edges_of x -defined the_Vertices_of x -valued Function-like V39( the_Edges_of x) V40( the_Edges_of x, the_Vertices_of x) Element of bool [:(the_Edges_of x),(the_Vertices_of x):]
the_Vertices_of x is non empty set
x . VertexSelector is set
[:(the_Edges_of x),(the_Vertices_of x):] is Relation-like set
bool [:(the_Edges_of x),(the_Vertices_of x):] is non empty set
x . TargetSelector is set
(the_Target_of x) . n is set
the_Source_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
G . SourceSelector is set
(the_Source_of G) . n is set
the_Source_of x is Relation-like the_Edges_of x -defined the_Vertices_of x -valued Function-like V39( the_Edges_of x) V40( the_Edges_of x, the_Vertices_of x) Element of bool [:(the_Edges_of x),(the_Vertices_of x):]
x . SourceSelector is set
(the_Source_of x) . n is set
n is set
H is set
the_Vertices_of x is non empty Element of bool (the_Vertices_of G)
the_Vertices_of x is non empty Element of bool (the_Vertices_of G)
x . VertexSelector is set
n is set
the_Source_of x is Relation-like the_Edges_of x -defined the_Vertices_of x -valued Function-like V39( the_Edges_of x) V40( the_Edges_of x, the_Vertices_of x) Element of bool [:(the_Edges_of x),(the_Vertices_of x):]
the_Edges_of x is set
the_Vertices_of x is non empty set
[:(the_Edges_of x),(the_Vertices_of x):] is Relation-like set
bool [:(the_Edges_of x),(the_Vertices_of x):] is non empty set
x . SourceSelector is set
(the_Source_of x) . n is set
(the_Source_of G) . n is set
(the_Source_of x) . n is set
the_Target_of x is Relation-like the_Edges_of x -defined the_Vertices_of x -valued Function-like V39( the_Edges_of x) V40( the_Edges_of x, the_Vertices_of x) Element of bool [:(the_Edges_of x),(the_Vertices_of x):]
x . TargetSelector is set
(the_Target_of x) . n is set
(the_Target_of G) . n is set
(the_Target_of x) . n is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
P is non empty Element of bool (the_Vertices_of G)
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
G .edgesBetween P is Element of bool (the_Edges_of G)
G .edgesInto P is Element of bool (the_Edges_of G)
G .edgesOutOf P is Element of bool (the_Edges_of G)
(G .edgesInto P) /\ (G .edgesOutOf P) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
x .edgesBetween P is Element of bool (the_Edges_of x)
the_Edges_of x is set
x . EdgeSelector is set
bool (the_Edges_of x) is non empty set
x .edgesInto P is Element of bool (the_Edges_of x)
x .edgesOutOf P is Element of bool (the_Edges_of x)
(x .edgesInto P) /\ (x .edgesOutOf P) is Element of bool (the_Edges_of x)
the_Edges_of x is Element of bool (the_Edges_of G)
x is set
the_Source_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
[:(the_Edges_of G),(the_Vertices_of G):] is Relation-like set
bool [:(the_Edges_of G),(the_Vertices_of G):] is non empty set
G . SourceSelector is set
(the_Source_of G) . x is set
the_Target_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
G . TargetSelector is set
(the_Target_of G) . x is set
the_Target_of x is Relation-like the_Edges_of x -defined the_Vertices_of x -valued Function-like V39( the_Edges_of x) V40( the_Edges_of x, the_Vertices_of x) Element of bool [:(the_Edges_of x),(the_Vertices_of x):]
the_Vertices_of x is non empty set
x . VertexSelector is set
[:(the_Edges_of x),(the_Vertices_of x):] is Relation-like set
bool [:(the_Edges_of x),(the_Vertices_of x):] is non empty set
x . TargetSelector is set
(the_Target_of x) . x is set
the_Source_of x is Relation-like the_Edges_of x -defined the_Vertices_of x -valued Function-like V39( the_Edges_of x) V40( the_Edges_of x, the_Vertices_of x) Element of bool [:(the_Edges_of x),(the_Vertices_of x):]
x . SourceSelector is set
(the_Source_of x) . x is set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
S + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P is Relation-like NAT -defined Function-like finite [Graph-like] finite set
P .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of P is non empty finite set
P . VertexSelector is set
len (the_Vertices_of P) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
P is Relation-like NAT -defined Function-like finite [Graph-like] finite set
P .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of P is non empty finite set
P . VertexSelector is set
len (the_Vertices_of P) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
S is Relation-like NAT -defined Function-like finite [Graph-like] finite set
S .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of S is non empty finite set
S . VertexSelector is set
len (the_Vertices_of S) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
G .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of G is non empty finite set
G . VertexSelector is set
len (the_Vertices_of G) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
S is Relation-like NAT -defined Function-like finite [Graph-like] finite set
S .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of S is non empty finite set
S . VertexSelector is set
len (the_Vertices_of S) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . P is set
S . x is set
S .last() is Element of the_Vertices_of G
S . (len S) is set
S .first() is Element of the_Vertices_of G
S . 1 is set
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
S . (len S) is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S .last() is Element of the_Vertices_of G
S . (len S) is set
S .first() is Element of the_Vertices_of G
S . 1 is set
P is set
S .addEdge P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .last()) .adj P is Element of the_Vertices_of G
G .walkOf ((S .last()),P,((S .last()) .adj P)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
S .append (G .walkOf ((S .last()),P,((S .last()) .adj P))) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(S .edgeSeq())) is finite Element of bool (the_Edges_of G)
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is Element of the_Vertices_of G
S . n is set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (n + 1) is set
x is Element of the_Vertices_of G
S . (n + 2) is set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .addEdge P) .last() is Element of the_Vertices_of G
len (S .addEdge P) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .addEdge P) . (len (S .addEdge P)) is set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len S) + (2 * 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) + 2) - 2 is complex V25() integer ext-real set
(S .addEdge P) . x is set
(S .addEdge P) . n is set
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
S . x is set
S . n is set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .last()) .edgesInOut() is Element of bool (the_Edges_of G)
{(S .last())} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
G .edgesInOut {(S .last())} is Element of bool (the_Edges_of G)
G .edgesInto {(S .last())} is Element of bool (the_Edges_of G)
G .edgesOutOf {(S .last())} is Element of bool (the_Edges_of G)
(G .edgesInto {(S .last())}) \/ (G .edgesOutOf {(S .last())}) is Element of bool (the_Edges_of G)
(S .addEdge P) .first() is Element of the_Vertices_of G
(S .addEdge P) . 1 is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .first() is Element of the_Vertices_of G
S . 1 is set
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Subwalk of S
P .first() is Element of the_Vertices_of G
P . 1 is set
P .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
dom P is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
len P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len (S .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (len (S .edgeSeq())) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (len (S .edgeSeq()))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
P . x is set
x div 2 is complex V25() integer ext-real set
(P .edgeSeq()) . (x div 2) is set
S . x is set
P . x is set
S . x is set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
- 2 is complex V25() integer ext-real non positive set
3 + (- 2) is complex V25() integer ext-real set
x + (- 2) is complex V25() integer ext-real set
- 1 is complex V25() integer ext-real non positive set
3 + (- 1) is complex V25() integer ext-real set
x + (- 1) is complex V25() integer ext-real set
x - 1 is complex V25() integer ext-real set
n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
P . n is set
S . n is set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x - (2 * 1) is non empty complex V25() integer non even ext-real set
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . H is set
H + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (H + 2) is set
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (H + 1) is set
S . H is set
S . (H + 2) is set
S . (H + 1) is set
P . x is set
S . x is set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . x is set
S . x is set
x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
P . x is set
S . x is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Subwalk of S
len P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (P .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (len (P .edgeSeq())) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (len (P .edgeSeq()))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P .first() is Element of the_Vertices_of G
P . 1 is set
S .first() is Element of the_Vertices_of G
S . 1 is set
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (S .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (len (S .edgeSeq())) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (len (S .edgeSeq()))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
bool (S .edgeSeq()) is non empty finite V32() set
x is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSubsequence-like Element of bool (S .edgeSeq())
Seq x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (len S) is set
len the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . P is set
S . x is set
S .cut (1,P) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .cut (x,(len S)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .cut (1,P)) .append (S .cut (x,(len S))) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len (S .cut (x,(len S))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (x,(len S)))) + x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len (S .cut (x,(len S)))) + x) - x is non empty complex V25() integer non even ext-real set
(len S) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) + 1) - x is non empty complex V25() integer non even ext-real set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len (S .cut (1,P)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (1,P))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len (S .cut (1,P))) + 1) - 1 is non empty complex V25() integer non even ext-real set
P + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(P + 1) - 1 is non empty complex V25() integer non even ext-real set
(S .cut (x,(len S))) .first() is Element of the_Vertices_of G
(S .cut (x,(len S))) . 1 is set
(S .cut (1,P)) .last() is Element of the_Vertices_of G
(S .cut (1,P)) . (len (S .cut (1,P))) is set
len ((S .cut (1,P)) .append (S .cut (x,(len S)))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len ((S .cut (1,P)) .append (S .cut (x,(len S))))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (1,P))) + (len (S .cut (x,(len S)))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) - x is complex V25() integer even ext-real set
((len S) - x) + P is non empty complex V25() integer non even ext-real set
((len S) - x) + x is non empty complex V25() integer non even ext-real set
(S .cut (x,(len S))) .last() is Element of the_Vertices_of G
(S .cut (x,(len S))) . (len (S .cut (x,(len S)))) is set
S . (len S) is set
(S .cut (1,P)) .first() is Element of the_Vertices_of G
(S .cut (1,P)) . 1 is set
S . ((2 * 0) + 1) is set
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
S .cut (1,x) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .cut (P,(len S)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .cut (1,x)) .append (S .cut (P,(len S))) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len (S .cut (P,(len S))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (P,(len S)))) + P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len (S .cut (P,(len S)))) + P) - P is non empty complex V25() integer non even ext-real set
(len S) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) + 1) - P is non empty complex V25() integer non even ext-real set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len (S .cut (1,x)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (1,x))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len (S .cut (1,x))) + 1) - 1 is non empty complex V25() integer non even ext-real set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x + 1) - 1 is non empty complex V25() integer non even ext-real set
(S .cut (P,(len S))) .first() is Element of the_Vertices_of G
(S .cut (P,(len S))) . 1 is set
(S .cut (1,x)) .last() is Element of the_Vertices_of G
(S .cut (1,x)) . (len (S .cut (1,x))) is set
len ((S .cut (1,x)) .append (S .cut (P,(len S)))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len ((S .cut (1,x)) .append (S .cut (P,(len S))))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (1,x))) + (len (S .cut (P,(len S)))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) - P is complex V25() integer even ext-real set
((len S) - P) + x is non empty complex V25() integer non even ext-real set
((len S) - P) + P is non empty complex V25() integer non even ext-real set
(S .cut (P,(len S))) .last() is Element of the_Vertices_of G
(S .cut (P,(len S))) . (len (S .cut (P,(len S)))) is set
S . (len S) is set
(S .cut (1,x)) .first() is Element of the_Vertices_of G
(S .cut (1,x)) . 1 is set
S . ((2 * 0) + 1) is set
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (len S) is set
P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of P
P .last() is Element of the_Vertices_of G
len P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (len P) is set
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of P .last() is Element of the_Vertices_of G
len the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of P . (len the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of P) is set
P .first() is Element of the_Vertices_of G
P . 1 is set
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of P .first() is Element of the_Vertices_of G
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of P . 1 is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (len S) is set
{ (len b₁) where b₁ is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G : b₁ is_Walk_from S .first() ,S .last() } is set
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S .last() is Element of the_Vertices_of G
len the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S . (len the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S) is set
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S .first() is Element of the_Vertices_of G
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of S . 1 is set
x is set
n is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V148() Element of bool NAT
min x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative set
n is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n .first() is Element of the_Vertices_of G
n . 1 is set
n .last() is Element of the_Vertices_of G
n . (len n) is set
H is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H .last() is Element of the_Vertices_of G
H . (len H) is set
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of H is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of H
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of H .first() is Element of the_Vertices_of G
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of H . 1 is set
H .first() is Element of the_Vertices_of G
H . 1 is set
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of H .last() is Element of the_Vertices_of G
len the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of H . (len the Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of H) is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . P is set
S . x is set
(len S) - x is complex V25() integer even ext-real set
((len S) - x) + x is non empty complex V25() integer non even ext-real set
((len S) - x) + (P + 2) is non empty complex V25() integer non even ext-real set
P + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S .cut (x,(len S)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .cut (x,(len S))) .last() is Element of the_Vertices_of G
len (S .cut (x,(len S))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .cut (x,(len S))) . (len (S .cut (x,(len S)))) is set
S .last() is Element of the_Vertices_of G
S . (len S) is set
S .cut (1,P) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
H is set
(S .cut (1,P)) .addEdge H is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .cut (1,P)) .last() is Element of the_Vertices_of G
len (S .cut (1,P)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .cut (1,P)) . (len (S .cut (1,P))) is set
((S .cut (1,P)) .last()) .adj H is Element of the_Vertices_of G
G .walkOf (((S .cut (1,P)) .last()),H,(((S .cut (1,P)) .last()) .adj H)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
(S .cut (1,P)) .append (G .walkOf (((S .cut (1,P)) .last()),H,(((S .cut (1,P)) .last()) .adj H))) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
((S .cut (1,P)) .addEdge H) .append (S .cut (x,(len S))) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (1,P))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((S .cut (1,P)) .addEdge H) .last() is Element of the_Vertices_of G
len ((S .cut (1,P)) .addEdge H) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((S .cut (1,P)) .addEdge H) . (len ((S .cut (1,P)) .addEdge H)) is set
(S .cut (1,P)) .first() is Element of the_Vertices_of G
(S .cut (1,P)) . 1 is set
S .first() is Element of the_Vertices_of G
S . 1 is set
((S .cut (1,P)) .addEdge H) .first() is Element of the_Vertices_of G
((S .cut (1,P)) .addEdge H) . 1 is set
(S .cut (x,(len S))) .first() is Element of the_Vertices_of G
(S .cut (x,(len S))) . 1 is set
len (((S .cut (1,P)) .addEdge H) .append (S .cut (x,(len S)))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (((S .cut (1,P)) .addEdge H) .append (S .cut (x,(len S))))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len ((S .cut (1,P)) .addEdge H)) + (len (S .cut (x,(len S)))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (x,(len S)))) + x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
the_Vertices_of P is non empty set
P . VertexSelector is set
the_Edges_of P is set
P . EdgeSelector is set
(the_Vertices_of P) \/ (the_Edges_of P) is non empty set
x is Relation-like NAT -defined (the_Vertices_of P) \/ (the_Edges_of P) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of P
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of P is non empty Element of bool (the_Vertices_of G)
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . n is set
x . x is set
H is set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . x is set
x . n is set
H is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S .cut (P,x) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .cut (P,x)) .first() is Element of the_Vertices_of G
(S .cut (P,x)) . 1 is set
(S .cut (P,x)) .last() is Element of the_Vertices_of G
len (S .cut (P,x)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .cut (P,x)) . (len (S .cut (P,x))) is set
n is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S .cut (1,P) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S .cut (x,(len S)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .cut (x,(len S))) .first() is Element of the_Vertices_of G
(S .cut (x,(len S))) . 1 is set
S . x is set
(S .cut (1,P)) .append n is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .cut (1,P)) .last() is Element of the_Vertices_of G
len (S .cut (1,P)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .cut (1,P)) . (len (S .cut (1,P))) is set
S . P is set
n .first() is Element of the_Vertices_of G
n . 1 is set
len ((S .cut (1,P)) .append n) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len ((S .cut (1,P)) .append n)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (1,P))) + (len n) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (1,P))) + (len (S .cut (P,x))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((S .cut (1,P)) .append n) .append (S .cut (x,(len S))) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
n .last() is Element of the_Vertices_of G
n . (len n) is set
((S .cut (1,P)) .append n) .last() is Element of the_Vertices_of G
((S .cut (1,P)) .append n) . (len ((S .cut (1,P)) .append n)) is set
len (((S .cut (1,P)) .append n) .append (S .cut (x,(len S)))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (((S .cut (1,P)) .append n) .append (S .cut (x,(len S))))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len (S .cut (x,(len S))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len ((S .cut (1,P)) .append n)) + (len (S .cut (x,(len S)))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (1,P))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (S .cut (P,x))) + P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x + 1) - 1 is non empty complex V25() integer non even ext-real set
((len ((S .cut (1,P)) .append n)) + 1) - 1 is non empty complex V25() integer non even ext-real set
(len (S .cut (x,(len S)))) + x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) + 1) - 1 is non empty complex V25() integer non even ext-real set
((len (((S .cut (1,P)) .append n) .append (S .cut (x,(len S))))) + 1) - 1 is non empty complex V25() integer non even ext-real set
(S .cut (x,(len S))) .last() is Element of the_Vertices_of G
(S .cut (x,(len S))) . (len (S .cut (x,(len S)))) is set
S . (len S) is set
(((S .cut (1,P)) .append n) .append (S .cut (x,(len S)))) .last() is Element of the_Vertices_of G
(((S .cut (1,P)) .append n) .append (S .cut (x,(len S)))) . (len (((S .cut (1,P)) .append n) .append (S .cut (x,(len S))))) is set
S .last() is Element of the_Vertices_of G
(S .cut (1,P)) .first() is Element of the_Vertices_of G
(S .cut (1,P)) . 1 is set
S . 1 is set
((S .cut (1,P)) .append n) .first() is Element of the_Vertices_of G
((S .cut (1,P)) .append n) . 1 is set
(((S .cut (1,P)) .append n) .append (S .cut (x,(len S)))) .first() is Element of the_Vertices_of G
(((S .cut (1,P)) .append n) .append (S .cut (x,(len S)))) . 1 is set
S .first() is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is non empty Element of bool (the_Vertices_of G)
P is non empty Element of bool (the_Vertices_of G)
{ (len b₁) where b₁ is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G : ( b₁ .first() in S & b₁ .last() in P ) } is set
x is set
n is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n .first() is Element of the_Vertices_of G
n . 1 is set
n .last() is Element of the_Vertices_of G
n . (len n) is set
x is set
H is set
y is Element of the_Vertices_of G
n is Element of the_Vertices_of G
C is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
C .first() is Element of the_Vertices_of G
C . 1 is set
C .last() is Element of the_Vertices_of G
len C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C . (len C) is set
C is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
C .last() is Element of the_Vertices_of G
len C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C . (len C) is set
C .first() is Element of the_Vertices_of G
C . 1 is set
a is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V148() Element of bool NAT
min a is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative set
b is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
b .first() is Element of the_Vertices_of G
b . 1 is set
b .last() is Element of the_Vertices_of G
b . (len b) is set
v is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
v .last() is Element of the_Vertices_of G
len v is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
v . (len v) is set
v .first() is Element of the_Vertices_of G
v . 1 is set
v is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
b . v is set
n1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
b .cut (n1,(len b)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
bg is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
bg .last() is Element of the_Vertices_of G
len bg is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
bg . (len bg) is set
(len bg) + n1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len b) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- v is complex V25() integer ext-real non positive set
((len b) + 1) + (- v) is complex V25() integer ext-real set
- 1 is complex V25() integer ext-real non positive set
v + (- 1) is complex V25() integer ext-real set
(((len b) + 1) + (- v)) + (v + (- 1)) is complex V25() integer ext-real set
(len bg) + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
bg .first() is Element of the_Vertices_of G
bg . 1 is set
n1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
b .cut (1,n1) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
bg is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
bg .last() is Element of the_Vertices_of G
len bg is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
bg . (len bg) is set
b . n1 is set
(len bg) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
bg .first() is Element of the_Vertices_of G
bg . 1 is set
S /\ P is Element of bool (the_Vertices_of G)
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
n is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
S is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Vertices_of S is non empty set
S . VertexSelector is set
P is Element of the_Vertices_of G
x is Element of the_Vertices_of G
x is set
n is Element of the_Vertices_of S
H is Element of the_Vertices_of S
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
the_Vertices_of P is non empty set
P . VertexSelector is set
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
n is Element of the_Vertices_of P
H is Element of the_Vertices_of P
the_Vertices_of P is non empty Element of bool (the_Vertices_of G)
y is set
y is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (len S) is set
S .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (S .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (1 + 2) is set
S . (1 + 1) is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is Element of the_Vertices_of G
{S,P,x} is non empty finite set
x is set
G .walkOf (S,x,P) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
(G .walkOf (S,x,P)) .vertices() is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
(G .walkOf (S,x,P)) .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),((G .walkOf (S,x,P)) .vertexSeq())) is finite Element of bool (the_Vertices_of G)
{S,P} is non empty finite Element of bool (the_Vertices_of G)
H is set
(G .walkOf (S,x,P)) .addEdge H is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(G .walkOf (S,x,P)) .last() is Element of the_Vertices_of G
len (G .walkOf (S,x,P)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(G .walkOf (S,x,P)) . (len (G .walkOf (S,x,P))) is set
((G .walkOf (S,x,P)) .last()) .adj H is Element of the_Vertices_of G
G .walkOf (((G .walkOf (S,x,P)) .last()),H,(((G .walkOf (S,x,P)) .last()) .adj H)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
(G .walkOf (S,x,P)) .append (G .walkOf (((G .walkOf (S,x,P)) .last()),H,(((G .walkOf (S,x,P)) .last()) .adj H))) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
((G .walkOf (S,x,P)) .addEdge H) .last() is Element of the_Vertices_of G
len ((G .walkOf (S,x,P)) .addEdge H) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((G .walkOf (S,x,P)) .addEdge H) . (len ((G .walkOf (S,x,P)) .addEdge H)) is set
(G .walkOf (S,x,P)) .first() is Element of the_Vertices_of G
(G .walkOf (S,x,P)) . 1 is set
((G .walkOf (S,x,P)) .addEdge H) .first() is Element of the_Vertices_of G
((G .walkOf (S,x,P)) .addEdge H) . 1 is set
C is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
C .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (C .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
C .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
K426((the_Edges_of G),(C .edgeSeq())) is finite Element of bool (the_Edges_of G)
C .vertices() is finite Element of bool (the_Vertices_of G)
C .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(C .vertexSeq())) is finite Element of bool (the_Vertices_of G)
C . 1 is set
C . 3 is set
C . 5 is set
{x,H} is non empty finite set
3 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * (C .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (C .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(G .walkOf (S,x,P)) .edges() is finite Element of bool (the_Edges_of G)
(G .walkOf (S,x,P)) .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),((G .walkOf (S,x,P)) .edgeSeq())) is finite Element of bool (the_Edges_of G)
{x} is non empty trivial finite 1 -element set
{H} is non empty trivial finite 1 -element set
{x} \/ {H} is non empty finite set
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
{S,P} \/ {x} is non empty finite Element of bool (the_Vertices_of G)
dom (G .walkOf (S,x,P)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
(G .walkOf (S,x,P)) . 3 is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
{S,P,x,x} is non empty finite set
{S,P,x} is non empty finite set
n is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
n .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (n .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
H is set
y is set
n .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
K426((the_Edges_of G),(n .edgeSeq())) is finite Element of bool (the_Edges_of G)
{H,y} is non empty finite set
n .vertices() is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
n .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(n .vertexSeq())) is finite Element of bool (the_Vertices_of G)
n . 1 is set
n . 3 is set
n . 5 is set
C is set
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n . C is set
n .addEdge C is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
n .last() is Element of the_Vertices_of G
n . (len n) is set
(n .last()) .adj C is Element of the_Vertices_of G
G .walkOf ((n .last()),C,((n .last()) .adj C)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
n .append (G .walkOf ((n .last()),C,((n .last()) .adj C))) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
a is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len a is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
a .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
a .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (a .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
a .vertices() is finite Element of bool (the_Vertices_of G)
a .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(a .vertexSeq())) is finite Element of bool (the_Vertices_of G)
a . 1 is set
a . 3 is set
a . 5 is set
a . 7 is set
5 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * (a .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (a .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
{S,P,x} \/ {x} is non empty finite set
dom n is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
a .last() is Element of the_Vertices_of G
a . (len a) is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is set
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in S & ex b₂ being Element of the_Vertices_of G st
( b₂ in S & (G,b₁,b₂) ) ) } is set
bool (the_Vertices_of G) is non empty set
x is set
x is Element of the_Vertices_of G
n is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
P is set
S is set
(G,S) is Element of bool (the_Vertices_of G)
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in S & ex b₂ being Element of the_Vertices_of G st
( b₂ in S & (G,b₁,b₂) ) ) } is set
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is set
(G,S) is Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in S & ex b₂ being Element of the_Vertices_of G st
( b₂ in S & (G,b₁,b₂) ) ) } is set
P is Element of the_Vertices_of G
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
S is Relation-like NAT -defined Function-like finite [Graph-like] set
P is set
(G,P) is Element of bool (the_Vertices_of G)
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in P & ex b₂ being Element of the_Vertices_of G st
( b₂ in P & (G,b₁,b₂) ) ) } is set
(S,P) is Element of bool (the_Vertices_of S)
the_Vertices_of S is non empty set
S . VertexSelector is set
bool (the_Vertices_of S) is non empty set
{ b₁ where b₁ is Element of the_Vertices_of S : ( not b₁ in P & ex b₂ being Element of the_Vertices_of S st
( b₂ in P & (S,b₁,b₂) ) ) } is set
x is set
x is Element of the_Vertices_of S
n is Element of the_Vertices_of S
H is Element of the_Vertices_of G
y is Element of the_Vertices_of G
x is set
x is Element of the_Vertices_of G
n is Element of the_Vertices_of G
H is Element of the_Vertices_of S
y is Element of the_Vertices_of S
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
{P} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
(G,{P}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {P} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {P} & (G,b₁,b₂) ) ) } is set
x is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
S is set
P is set
{P} is non empty trivial finite 1 -element set
(G,{P}) is Element of bool (the_Vertices_of G)
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {P} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {P} & (G,b₁,b₂) ) ) } is set
{S} is non empty trivial finite 1 -element set
(G,{S}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {S} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {S} & (G,b₁,b₂) ) ) } is set
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
S .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (S .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S .vertices() is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
S .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(S .vertexSeq())) is finite Element of bool (the_Vertices_of G)
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) - 2 is complex V25() integer ext-real set
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 4 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (S .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
8 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(2 * (S .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
9 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P is Element of the_Vertices_of G
{P} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
(G,{P}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {P} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {P} & (G,b₁,b₂) ) ) } is set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . x is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- 6 is complex V25() integer ext-real non positive set
9 + (- 6) is complex V25() integer ext-real set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . x is set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . n is set
S .last() is Element of the_Vertices_of G
S . (len S) is set
S .first() is Element of the_Vertices_of G
S . 1 is set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . n is set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . n is set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . n is set
H is Element of the_Vertices_of G
- 2 is complex V25() integer ext-real non positive set
9 + (- 2) is complex V25() integer ext-real set
(len S) + (- 2) is complex V25() integer ext-real set
(len S) - (2 * 1) is non empty complex V25() integer non even ext-real set
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
y is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . y is set
y + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x + 2) + (- 2) is complex V25() integer ext-real set
- 8 is complex V25() integer ext-real non positive set
9 + (- 8) is complex V25() integer ext-real set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (n + 1) is set
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
S . (len S) is set
y + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (y + 2) is set
y + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (y + 2) is set
y + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (y + 2) is set
y + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (y + 2) is set
C is Element of the_Vertices_of G
y + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (y + 1) is set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- 2 is complex V25() integer ext-real non positive set
3 + (- 2) is complex V25() integer ext-real set
x + (- 2) is complex V25() integer ext-real set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x - (2 * 1) is non empty complex V25() integer non even ext-real set
x - 2 is complex V25() integer ext-real set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . x is set
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) - 2) + 2 is complex V25() integer ext-real set
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . H is set
x + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
y is Element of the_Vertices_of G
x + 4 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- 3 is complex V25() integer ext-real non positive set
9 + (- 3) is complex V25() integer ext-real set
(len S) + (- 2) is complex V25() integer ext-real set
n is Element of the_Vertices_of G
H + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x + 2) + (- 2) is complex V25() integer ext-real set
S . (x + 2) is set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (x + 1) is set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (x + 1) is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
S .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (S .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S .vertices() is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
S .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(S .vertexSeq())) is finite Element of bool (the_Vertices_of G)
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
K426((the_Edges_of G),(S .edgeSeq())) is finite Element of bool (the_Edges_of G)
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 4 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (S .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
8 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(2 * (S .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
9 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P is Element of the_Vertices_of G
{P} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
(G,{P}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {P} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {P} & (G,b₁,b₂) ) ) } is set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . x is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . x is set
S .last() is Element of the_Vertices_of G
S . (len S) is set
S .first() is Element of the_Vertices_of G
S . 1 is set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . x is set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . x is set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . x is set
- 2 is complex V25() integer ext-real non positive set
9 + (- 2) is complex V25() integer ext-real set
(len S) + (- 2) is complex V25() integer ext-real set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len S) - (2 * 1) is non empty complex V25() integer non even ext-real set
(len S) - 2 is complex V25() integer ext-real set
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
H + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . H is set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . n is set
(len S) + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(H + 2) + (- 2) is complex V25() integer ext-real set
- 6 is complex V25() integer ext-real non positive set
9 + (- 6) is complex V25() integer ext-real set
- 8 is complex V25() integer ext-real non positive set
9 + (- 8) is complex V25() integer ext-real set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (x + 1) is set
y is Element of the_Vertices_of G
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
S . (len S) is set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (n + 2) is set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (n + 2) is set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (n + 2) is set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (n + 2) is set
C is Element of the_Vertices_of G
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (n + 1) is set
C is set
a is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S . a is set
the_Source_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
[:(the_Edges_of G),(the_Vertices_of G):] is Relation-like set
bool [:(the_Edges_of G),(the_Vertices_of G):] is non empty set
G . SourceSelector is set
(the_Source_of G) . C is set
the_Target_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
G . TargetSelector is set
(the_Target_of G) . C is set
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
a - 1 is non empty complex V25() integer non even ext-real set
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (a + 1) is set
b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . b is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(a - 1) + (1 + 1) is complex V25() integer ext-real set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- 2 is complex V25() integer ext-real non positive set
3 + (- 2) is complex V25() integer ext-real set
x + (- 2) is complex V25() integer ext-real set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x - (2 * 1) is non empty complex V25() integer non even ext-real set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . x is set
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . H is set
x + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n is Element of the_Vertices_of G
x + 4 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x + 2) + (- 2) is complex V25() integer ext-real set
S . (x + 2) is set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (x + 1) is set
(len S) - 2 is complex V25() integer ext-real set
((len S) - 2) + 2 is complex V25() integer ext-real set
y is Element of the_Vertices_of G
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (x + 1) is set
C is set
C is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S . C is set
the_Source_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
[:(the_Edges_of G),(the_Vertices_of G):] is Relation-like set
bool [:(the_Edges_of G),(the_Vertices_of G):] is non empty set
G . SourceSelector is set
(the_Source_of G) . C is set
the_Target_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
G . TargetSelector is set
(the_Target_of G) . C is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C - 1 is non empty complex V25() integer non even ext-real set
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
S . (C + 1) is set
a is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . a is set
(C - 1) + (1 + 1) is complex V25() integer ext-real set
G is Relation-like NAT -defined Function-like finite [Graph-like] loopless set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
{S} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
(G,{S}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {S} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {S} & (G,b₁,b₂) ) ) } is set
S .edgesInOut() is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInOut {S} is Element of bool (the_Edges_of G)
G .edgesInto {S} is Element of bool (the_Edges_of G)
G .edgesOutOf {S} is Element of bool (the_Edges_of G)
(G .edgesInto {S}) \/ (G .edgesOutOf {S}) is Element of bool (the_Edges_of G)
P is set
x is Element of the_Vertices_of G
the_Target_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
[:(the_Edges_of G),(the_Vertices_of G):] is Relation-like set
bool [:(the_Edges_of G),(the_Vertices_of G):] is non empty set
G . TargetSelector is set
(the_Target_of G) . P is set
the_Source_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
G . SourceSelector is set
(the_Source_of G) . P is set
P is set
x is Element of the_Vertices_of G
x is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
P is non empty Element of bool (the_Vertices_of G)
G .edgesBetween P is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto P is Element of bool (the_Edges_of G)
G .edgesOutOf P is Element of bool (the_Edges_of G)
(G .edgesInto P) /\ (G .edgesOutOf P) is Element of bool (the_Edges_of G)
x is Element of the_Vertices_of G
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
P \/ {x} is non empty Element of bool (the_Vertices_of G)
G .edgesBetween (P \/ {x}) is Element of bool (the_Edges_of G)
G .edgesInto (P \/ {x}) is Element of bool (the_Edges_of G)
G .edgesOutOf (P \/ {x}) is Element of bool (the_Edges_of G)
(G .edgesInto (P \/ {x})) /\ (G .edgesOutOf (P \/ {x})) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,P,G .edgesBetween P
the_Vertices_of x is non empty Element of bool (the_Vertices_of G)
x . VertexSelector is set
(G,(the_Vertices_of x)) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in the_Vertices_of x & ex b₂ being Element of the_Vertices_of G st
( b₂ in the_Vertices_of x & (G,b₁,b₂) ) ) } is set
n is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,P \/ {x},G .edgesBetween (P \/ {x})
H is Element of the_Vertices_of G
y is set
the_Vertices_of n is non empty set
n . VertexSelector is set
the_Edges_of n is set
n . EdgeSelector is set
(the_Vertices_of n) \/ (the_Edges_of n) is non empty set
C is Element of the_Vertices_of n
a is Element of the_Vertices_of n
the_Vertices_of n is non empty Element of bool (the_Vertices_of G)
C is Element of bool (the_Vertices_of G)
n .edgesBetween P is Element of bool (the_Edges_of n)
bool (the_Edges_of n) is non empty set
n .edgesInto P is Element of bool (the_Edges_of n)
n .edgesOutOf P is Element of bool (the_Edges_of n)
(n .edgesInto P) /\ (n .edgesOutOf P) is Element of bool (the_Edges_of n)
the_Vertices_of x is non empty set
the_Edges_of x is set
x . EdgeSelector is set
(the_Vertices_of x) \/ (the_Edges_of x) is non empty set
b is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
v is Relation-like NAT -defined (the_Vertices_of n) \/ (the_Edges_of n) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of n
the_Vertices_of x is non empty set
the_Edges_of x is set
x . EdgeSelector is set
(the_Vertices_of x) \/ (the_Edges_of x) is non empty set
b is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
v is Relation-like NAT -defined (the_Vertices_of n) \/ (the_Edges_of n) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of n
n .walkOf (H,y,x) is Relation-like NAT -defined (the_Vertices_of n) \/ (the_Edges_of n) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of n
v .append (n .walkOf (H,y,x)) is Relation-like NAT -defined (the_Vertices_of n) \/ (the_Edges_of n) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of n
the_Vertices_of x is non empty set
the_Edges_of x is set
x . EdgeSelector is set
(the_Vertices_of x) \/ (the_Edges_of x) is non empty set
b is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
n .walkOf (x,y,H) is Relation-like NAT -defined (the_Vertices_of n) \/ (the_Edges_of n) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of n
v is Relation-like NAT -defined (the_Vertices_of n) \/ (the_Edges_of n) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of n
(n .walkOf (x,y,H)) .append v is Relation-like NAT -defined (the_Vertices_of n) \/ (the_Edges_of n) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of n
n .walkOf C is Relation-like NAT -defined (the_Vertices_of n) \/ (the_Edges_of n) -valued Function-like finite FinSequence-like FinSubsequence-like closed V106(n) trivial Trail-like Path-like vertex-distinct Walk of n
<*C*> is non empty trivial Relation-like NAT -defined the_Vertices_of n -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the_Vertices_of n
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
the_Vertices_of P is non empty set
P . VertexSelector is set
x is Element of the_Vertices_of G
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
(G,{x}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {x} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {x} & (G,b₁,b₂) ) ) } is set
x is Element of the_Vertices_of P
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of P)
bool (the_Vertices_of P) is non empty set
(P,{x}) is Element of bool (the_Vertices_of P)
{ b₁ where b₁ is Element of the_Vertices_of P : ( not b₁ in {x} & ex b₂ being Element of the_Vertices_of P st
( b₂ in {x} & (P,b₁,b₂) ) ) } is set
n is set
H is Element of the_Vertices_of G
C is set
y is Element of the_Vertices_of P
the_Vertices_of P is non empty Element of bool (the_Vertices_of G)
H is Element of the_Vertices_of P
C is set
y is Element of the_Vertices_of G
S is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
(G,S) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in S & ex b₂ being Element of the_Vertices_of G st
( b₂ in S & (G,b₁,b₂) ) ) } is set
G .edgesBetween (G,S) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto (G,S) is Element of bool (the_Edges_of G)
G .edgesOutOf (G,S) is Element of bool (the_Edges_of G)
(G .edgesInto (G,S)) /\ (G .edgesOutOf (G,S)) is Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(G,S),G .edgesBetween (G,S) is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(G,S),G .edgesBetween (G,S)
G is Relation-like NAT -defined Function-like finite [Graph-like] set
S is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Vertices_of S is non empty set
S . VertexSelector is set
P is Element of the_Vertices_of G
{P} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
x is Element of the_Vertices_of S
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of S)
bool (the_Vertices_of S) is non empty set
(S,{x}) is Element of bool (the_Vertices_of S)
{ b₁ where b₁ is Element of the_Vertices_of S : ( not b₁ in {x} & ex b₂ being Element of the_Vertices_of S st
( b₂ in {x} & (S,b₁,b₂) ) ) } is set
(G,{P}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {P} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {P} & (G,b₁,b₂) ) ) } is set
H is Relation-like NAT -defined Function-like finite [Graph-like] (G,{P})
y is Relation-like NAT -defined Function-like finite [Graph-like] (S,{x})
G .edgesBetween (G,{P}) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto (G,{P}) is Element of bool (the_Edges_of G)
G .edgesOutOf (G,{P}) is Element of bool (the_Edges_of G)
(G .edgesInto (G,{P})) /\ (G .edgesOutOf (G,{P})) is Element of bool (the_Edges_of G)
S .edgesBetween (S,{x}) is Element of bool (the_Edges_of S)
the_Edges_of S is set
S . EdgeSelector is set
bool (the_Edges_of S) is non empty set
S .edgesInto (S,{x}) is Element of bool (the_Edges_of S)
S .edgesOutOf (S,{x}) is Element of bool (the_Edges_of S)
(S .edgesInto (S,{x})) /\ (S .edgesOutOf (S,{x})) is Element of bool (the_Edges_of S)
the_Vertices_of H is non empty Element of bool (the_Vertices_of G)
H . VertexSelector is set
the_Vertices_of y is non empty Element of bool (the_Vertices_of S)
y . VertexSelector is set
S .edgesBetween (G,{P}) is Element of bool (the_Edges_of S)
S .edgesInto (G,{P}) is Element of bool (the_Edges_of S)
S .edgesOutOf (G,{P}) is Element of bool (the_Edges_of S)
(S .edgesInto (G,{P})) /\ (S .edgesOutOf (G,{P})) is Element of bool (the_Edges_of S)
the_Edges_of H is Element of bool (the_Edges_of G)
H . EdgeSelector is set
the_Edges_of y is Element of bool (the_Edges_of S)
y . EdgeSelector is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
the_Vertices_of P is non empty set
P . VertexSelector is set
x is Element of the_Vertices_of G
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
(G,{x}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {x} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {x} & (G,b₁,b₂) ) ) } is set
x is Element of the_Vertices_of P
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of P)
bool (the_Vertices_of P) is non empty set
(P,{x}) is Element of bool (the_Vertices_of P)
{ b₁ where b₁ is Element of the_Vertices_of P : ( not b₁ in {x} & ex b₂ being Element of the_Vertices_of P st
( b₂ in {x} & (P,b₁,b₂) ) ) } is set
n is Relation-like NAT -defined Function-like finite [Graph-like] (G,{x})
H is Relation-like NAT -defined Function-like finite [Graph-like] (P,{x})
P .edgesBetween (P,{x}) is Element of bool (the_Edges_of P)
the_Edges_of P is set
P . EdgeSelector is set
bool (the_Edges_of P) is non empty set
P .edgesInto (P,{x}) is Element of bool (the_Edges_of P)
P .edgesOutOf (P,{x}) is Element of bool (the_Edges_of P)
(P .edgesInto (P,{x})) /\ (P .edgesOutOf (P,{x})) is Element of bool (the_Edges_of P)
G .edgesBetween (G,{x}) is Element of bool (the_Edges_of G)
G .edgesInto (G,{x}) is Element of bool (the_Edges_of G)
G .edgesOutOf (G,{x}) is Element of bool (the_Edges_of G)
(G .edgesInto (G,{x})) /\ (G .edgesOutOf (G,{x})) is Element of bool (the_Edges_of G)
the_Edges_of n is Element of bool (the_Edges_of G)
n . EdgeSelector is set
the_Vertices_of n is non empty Element of bool (the_Vertices_of G)
n . VertexSelector is set
the_Vertices_of H is non empty Element of bool (the_Vertices_of P)
H . VertexSelector is set
the_Edges_of n is set
the_Edges_of H is set
H . EdgeSelector is set
the_Vertices_of n is non empty set
the_Vertices_of H is non empty set
the_Source_of n is Relation-like the_Edges_of n -defined the_Vertices_of n -valued Function-like V39( the_Edges_of n) V40( the_Edges_of n, the_Vertices_of n) Element of bool [:(the_Edges_of n),(the_Vertices_of n):]
[:(the_Edges_of n),(the_Vertices_of n):] is Relation-like set
bool [:(the_Edges_of n),(the_Vertices_of n):] is non empty set
n . SourceSelector is set
the_Source_of H is Relation-like the_Edges_of H -defined the_Vertices_of H -valued Function-like V39( the_Edges_of H) V40( the_Edges_of H, the_Vertices_of H) Element of bool [:(the_Edges_of H),(the_Vertices_of H):]
[:(the_Edges_of H),(the_Vertices_of H):] is Relation-like set
bool [:(the_Edges_of H),(the_Vertices_of H):] is non empty set
H . SourceSelector is set
the_Target_of n is Relation-like the_Edges_of n -defined the_Vertices_of n -valued Function-like V39( the_Edges_of n) V40( the_Edges_of n, the_Vertices_of n) Element of bool [:(the_Edges_of n),(the_Vertices_of n):]
n . TargetSelector is set
the_Target_of H is Relation-like the_Edges_of H -defined the_Vertices_of H -valued Function-like V39( the_Edges_of H) V40( the_Edges_of H, the_Vertices_of H) Element of bool [:(the_Edges_of H),(the_Vertices_of H):]
H . TargetSelector is set
the_Edges_of H is Element of bool (the_Edges_of P)
the_Target_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
[:(the_Edges_of G),(the_Vertices_of G):] is Relation-like set
bool [:(the_Edges_of G),(the_Vertices_of G):] is non empty set
G . TargetSelector is set
(the_Target_of G) | (the_Edges_of n) is Relation-like the_Edges_of n -defined the_Edges_of G -defined the_Vertices_of G -valued Function-like set
P .edgesBetween (G,{x}) is Element of bool (the_Edges_of P)
P .edgesInto (G,{x}) is Element of bool (the_Edges_of P)
P .edgesOutOf (G,{x}) is Element of bool (the_Edges_of P)
(P .edgesInto (G,{x})) /\ (P .edgesOutOf (G,{x})) is Element of bool (the_Edges_of P)
the_Source_of G is Relation-like the_Edges_of G -defined the_Vertices_of G -valued Function-like V39( the_Edges_of G) V40( the_Edges_of G, the_Vertices_of G) Element of bool [:(the_Edges_of G),(the_Vertices_of G):]
G . SourceSelector is set
(the_Source_of G) | (the_Edges_of n) is Relation-like the_Edges_of n -defined the_Edges_of G -defined the_Vertices_of G -valued Function-like set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
{S} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
P is Element of the_Vertices_of G
x is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the Relation-like NAT -defined Function-like finite [Graph-like] finite loopless trivial non-multi non-Dmulti simple Dsimple connected acyclic V115() () set is Relation-like NAT -defined Function-like finite [Graph-like] finite loopless trivial non-multi non-Dmulti simple Dsimple connected acyclic V115() () set
{0,1} is non empty finite V32() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
{0} is non empty trivial functional finite V32() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
0 .--> 0 is Relation-like NAT -defined {0} -defined NAT -valued Function-like one-to-one finite Function-yielding V100() set
{0} is non empty trivial functional finite V32() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() set
{0} --> 0 is non empty Relation-like {0} -defined NAT -valued {0} -valued Function-like constant finite V39({0}) V40({0},{0}) Function-yielding V100() Element of bool [:{0},{0}:]
[:{0},{0}:] is non empty Relation-like finite set
bool [:{0},{0}:] is non empty finite V32() set
0 .--> 1 is Relation-like NAT -defined {0} -defined NAT -valued Function-like one-to-one finite set
{0} --> 1 is non empty Relation-like non-empty {0} -defined NAT -valued {1} -valued Function-like constant finite V39({0}) V40({0},{1}) Element of bool [:{0},{1}:]
{1} is non empty trivial finite V32() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() set
[:{0},{1}:] is non empty Relation-like finite set
bool [:{0},{1}:] is non empty finite V32() set
dom (0 .--> 1) is finite set
x is set
(0 .--> 1) . x is set
x is set
(0 .--> 0) . x is Relation-like Function-like set
[:{0},{0,1}:] is non empty Relation-like finite set
bool [:{0},{0,1}:] is non empty finite V32() set
dom (0 .--> 0) is finite set
n is non empty Relation-like {0} -defined {0,1} -valued Function-like finite V39({0}) V40({0},{0,1}) Element of bool [:{0},{0,1}:]
x is non empty Relation-like {0} -defined {0,1} -valued Function-like finite V39({0}) V40({0},{0,1}) Element of bool [:{0},{0,1}:]
createGraph ({0,1},{0},n,x) is Relation-like NAT -defined Function-like finite [Graph-like] finite set
<*{0,1},{0},n,x*> is set
the_Edges_of (createGraph ({0,1},{0},n,x)) is finite set
(createGraph ({0,1},{0},n,x)) . EdgeSelector is set
the_Vertices_of (createGraph ({0,1},{0},n,x)) is non empty finite set
(createGraph ({0,1},{0},n,x)) . VertexSelector is set
the_Source_of (createGraph ({0,1},{0},n,x)) is Relation-like the_Edges_of (createGraph ({0,1},{0},n,x)) -defined the_Vertices_of (createGraph ({0,1},{0},n,x)) -valued Function-like finite V39( the_Edges_of (createGraph ({0,1},{0},n,x))) V40( the_Edges_of (createGraph ({0,1},{0},n,x)), the_Vertices_of (createGraph ({0,1},{0},n,x))) Element of bool [:(the_Edges_of (createGraph ({0,1},{0},n,x))),(the_Vertices_of (createGraph ({0,1},{0},n,x))):]
[:(the_Edges_of (createGraph ({0,1},{0},n,x))),(the_Vertices_of (createGraph ({0,1},{0},n,x))):] is Relation-like finite set
bool [:(the_Edges_of (createGraph ({0,1},{0},n,x))),(the_Vertices_of (createGraph ({0,1},{0},n,x))):] is non empty finite V32() set
(createGraph ({0,1},{0},n,x)) . SourceSelector is set
(the_Source_of (createGraph ({0,1},{0},n,x))) . 0 is set
C is set
a is set
y is set
C is set
card (the_Vertices_of (createGraph ({0,1},{0},n,x))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of omega
y is set
{y} is non empty trivial finite 1 -element set
the_Target_of (createGraph ({0,1},{0},n,x)) is Relation-like the_Edges_of (createGraph ({0,1},{0},n,x)) -defined the_Vertices_of (createGraph ({0,1},{0},n,x)) -valued Function-like finite V39( the_Edges_of (createGraph ({0,1},{0},n,x))) V40( the_Edges_of (createGraph ({0,1},{0},n,x)), the_Vertices_of (createGraph ({0,1},{0},n,x))) Element of bool [:(the_Edges_of (createGraph ({0,1},{0},n,x))),(the_Vertices_of (createGraph ({0,1},{0},n,x))):]
(createGraph ({0,1},{0},n,x)) . TargetSelector is set
(the_Target_of (createGraph ({0,1},{0},n,x))) . 0 is set
y is set
C is set
C is Element of the_Vertices_of (createGraph ({0,1},{0},n,x))
y is Element of the_Vertices_of (createGraph ({0,1},{0},n,x))
C is Element of the_Vertices_of (createGraph ({0,1},{0},n,x))
G is Relation-like NAT -defined Function-like finite [Graph-like] set
S is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of S is non empty set
S . VertexSelector is set
P is Element of the_Vertices_of S
x is Element of the_Vertices_of S
the_Vertices_of G is non empty set
G . VertexSelector is set
n is Element of the_Vertices_of G
x is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] () set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
S is Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
the_Vertices_of P is non empty Element of bool (the_Vertices_of G)
P . VertexSelector is set
the_Vertices_of P is non empty set
x is Element of the_Vertices_of P
x is Element of the_Vertices_of P
H is Element of the_Vertices_of G
n is Element of the_Vertices_of G
y is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
G is Relation-like NAT -defined Function-like finite [Graph-like] () set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
{S} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
P is Relation-like NAT -defined Function-like finite [Graph-like] (G,{S})
(G,{S}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {S} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {S} & (G,b₁,b₂) ) ) } is set
G .edgesBetween (G,{S}) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto (G,{S}) is Element of bool (the_Edges_of G)
G .edgesOutOf (G,{S}) is Element of bool (the_Edges_of G)
(G .edgesInto (G,{S})) /\ (G .edgesOutOf (G,{S})) is Element of bool (the_Edges_of G)
G is Relation-like NAT -defined Function-like finite [Graph-like] trivial connected () set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is Element of the_Vertices_of G
{S} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty finite V32() set
(G,{S}) is finite Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {S} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {S} & (G,b₁,b₂) ) ) } is set
P is set
x is Element of the_Vertices_of G
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
x is Element of the_Vertices_of G
n is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
S is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Vertices_of S is non empty set
S . VertexSelector is set
P is Element of the_Vertices_of G
x is Element of the_Vertices_of S
{P} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
(G,{P}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {P} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {P} & (G,b₁,b₂) ) ) } is set
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of S)
bool (the_Vertices_of S) is non empty set
(S,{x}) is Element of bool (the_Vertices_of S)
{ b₁ where b₁ is Element of the_Vertices_of S : ( not b₁ in {x} & ex b₂ being Element of the_Vertices_of S st
( b₂ in {x} & (S,b₁,b₂) ) ) } is set
{P} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
(G,{P}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {P} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {P} & (G,b₁,b₂) ) ) } is set
the Relation-like NAT -defined Function-like finite [Graph-like] (G,{P}) is Relation-like NAT -defined Function-like finite [Graph-like] (G,{P})
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of S)
bool (the_Vertices_of S) is non empty set
n is Relation-like NAT -defined Function-like finite [Graph-like] (S,{x})
{P} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
(G,{P}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {P} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {P} & (G,b₁,b₂) ) ) } is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
the_Vertices_of P is non empty set
P . VertexSelector is set
x is Element of the_Vertices_of G
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
(G,{x}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {x} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {x} & (G,b₁,b₂) ) ) } is set
x is Element of the_Vertices_of P
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of P)
bool (the_Vertices_of P) is non empty set
(P,{x}) is Element of bool (the_Vertices_of P)
{ b₁ where b₁ is Element of the_Vertices_of P : ( not b₁ in {x} & ex b₂ being Element of the_Vertices_of P st
( b₂ in {x} & (P,b₁,b₂) ) ) } is set
the Relation-like NAT -defined Function-like finite [Graph-like] (G,{x}) is Relation-like NAT -defined Function-like finite [Graph-like] (G,{x})
H is Relation-like NAT -defined Function-like finite [Graph-like] (P,{x})
the Relation-like NAT -defined Function-like finite [Graph-like] (P,{x}) is Relation-like NAT -defined Function-like finite [Graph-like] (P,{x})
H is Relation-like NAT -defined Function-like finite [Graph-like] (G,{x})
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
{S} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
(G,{S}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {S} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {S} & (G,b₁,b₂) ) ) } is set
the Relation-like NAT -defined Function-like finite [Graph-like] (G,{S}) is Relation-like NAT -defined Function-like finite [Graph-like] (G,{S})
x is set
x is set
G .edgesBetween (G,{S}) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto (G,{S}) is Element of bool (the_Edges_of G)
G .edgesOutOf (G,{S}) is Element of bool (the_Edges_of G)
(G .edgesInto (G,{S})) /\ (G .edgesOutOf (G,{S})) is Element of bool (the_Edges_of G)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] (G,{S}) is non empty set
the Relation-like NAT -defined Function-like finite [Graph-like] (G,{S}) . VertexSelector is set
n is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] (G,{S})
H is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] (G,{S})
y is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
{S} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
(G,{S}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {S} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {S} & (G,b₁,b₂) ) ) } is set
{S} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
(G,{S}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {S} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {S} & (G,b₁,b₂) ) ) } is set
P is Relation-like NAT -defined Function-like finite [Graph-like] (G,{S})
G .edgesBetween (G,{S}) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto (G,{S}) is Element of bool (the_Edges_of G)
G .edgesOutOf (G,{S}) is Element of bool (the_Edges_of G)
(G .edgesInto (G,{S})) /\ (G .edgesOutOf (G,{S})) is Element of bool (the_Edges_of G)
the_Vertices_of P is non empty set
P . VertexSelector is set
x is Element of the_Vertices_of P
x is Element of the_Vertices_of P
the_Vertices_of P is non empty Element of bool (the_Vertices_of G)
H is Element of the_Vertices_of G
n is Element of the_Vertices_of G
y is Element of the_Vertices_of G
{S} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
(G,{S}) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {S} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {S} & (G,b₁,b₂) ) ) } is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
bool (the_Vertices_of G) is non empty set
{S,P} is non empty finite Element of bool (the_Vertices_of G)
(the_Vertices_of G) \ {S,P} is Element of bool (the_Vertices_of G)
x is set
(the_Vertices_of G) \ ((the_Vertices_of G) \ {S,P}) is Element of bool (the_Vertices_of G)
G .edgesBetween ((the_Vertices_of G) \ ((the_Vertices_of G) \ {S,P})) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto ((the_Vertices_of G) \ ((the_Vertices_of G) \ {S,P})) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ ((the_Vertices_of G) \ {S,P})) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ ((the_Vertices_of G) \ {S,P}))) /\ (G .edgesOutOf ((the_Vertices_of G) \ ((the_Vertices_of G) \ {S,P}))) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ ((the_Vertices_of G) \ {S,P}),G .edgesBetween ((the_Vertices_of G) \ ((the_Vertices_of G) \ {S,P}))
the_Vertices_of x is non empty set
x . VertexSelector is set
the_Edges_of x is set
x . EdgeSelector is set
(the_Vertices_of x) \/ (the_Edges_of x) is non empty set
n is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
H is set
n .first() is Element of the_Vertices_of x
n . 1 is set
n .vertices() is finite Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty set
n .vertexSeq() is Relation-like NAT -defined the_Vertices_of x -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of x
K426((the_Vertices_of x),(n .vertexSeq())) is finite Element of bool (the_Vertices_of x)
n .last() is Element of the_Vertices_of x
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n . (len n) is set
n .vertices() is finite Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty set
n .vertexSeq() is Relation-like NAT -defined the_Vertices_of x -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of x
K426((the_Vertices_of x),(n .vertexSeq())) is finite Element of bool (the_Vertices_of x)
n .vertices() is finite Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty set
n .vertexSeq() is Relation-like NAT -defined the_Vertices_of x -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of x
K426((the_Vertices_of x),(n .vertexSeq())) is finite Element of bool (the_Vertices_of x)
n .vertices() is finite Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty set
n .vertexSeq() is Relation-like NAT -defined the_Vertices_of x -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of x
K426((the_Vertices_of x),(n .vertexSeq())) is finite Element of bool (the_Vertices_of x)
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
y is set
H is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
H .vertices() is finite Element of bool (the_Vertices_of G)
H .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(H .vertexSeq())) is finite Element of bool (the_Vertices_of G)
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n . C is set
len H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H . C is set
the_Vertices_of x is non empty Element of bool (the_Vertices_of G)
(the_Vertices_of G) /\ {S,P} is finite Element of bool (the_Vertices_of G)
y is set
C is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
(the_Vertices_of G) \ x is Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
G .edgesBetween ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of x is non empty set
x . VertexSelector is set
the_Edges_of x is set
x . EdgeSelector is set
(the_Vertices_of x) \/ (the_Edges_of x) is non empty set
n is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
n .reverse() is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is Element of bool (the_Vertices_of G)
(the_Vertices_of G) \ x is Element of bool (the_Vertices_of G)
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
x .vertices() is finite Element of bool (the_Vertices_of G)
x .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(x .vertexSeq())) is finite Element of bool (the_Vertices_of G)
G .edgesBetween ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
G .edgesInto ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is Element of bool (the_Edges_of G)
n is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of n is non empty Element of bool (the_Vertices_of G)
n . VertexSelector is set
the_Edges_of n is Element of bool (the_Edges_of G)
n . EdgeSelector is set
G .edgesBetween (the_Vertices_of n) is Element of bool (the_Edges_of G)
G .edgesInto (the_Vertices_of n) is Element of bool (the_Edges_of G)
G .edgesOutOf (the_Vertices_of n) is Element of bool (the_Edges_of G)
(G .edgesInto (the_Vertices_of n)) /\ (G .edgesOutOf (the_Vertices_of n)) is Element of bool (the_Edges_of G)
x .edges() is finite Element of bool (the_Edges_of G)
x .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(x .edgeSeq())) is finite Element of bool (the_Edges_of G)
G .edgesBetween (x .vertices()) is Element of bool (the_Edges_of G)
G .edgesInto (x .vertices()) is Element of bool (the_Edges_of G)
G .edgesOutOf (x .vertices()) is Element of bool (the_Edges_of G)
(G .edgesInto (x .vertices())) /\ (G .edgesOutOf (x .vertices())) is Element of bool (the_Edges_of G)
H is set
the_Vertices_of n is non empty set
the_Edges_of n is set
(the_Vertices_of n) \/ (the_Edges_of n) is non empty set
x .last() is Element of the_Vertices_of G
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . (len x) is set
H is Relation-like NAT -defined (the_Vertices_of n) \/ (the_Edges_of n) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of n
H .last() is Element of the_Vertices_of n
len H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H . (len H) is set
x .first() is Element of the_Vertices_of G
x . 1 is set
H .first() is Element of the_Vertices_of n
H . 1 is set
n is Element of the_Vertices_of G
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of x is non empty set
x . VertexSelector is set
the_Edges_of x is set
x . EdgeSelector is set
(the_Vertices_of x) \/ (the_Edges_of x) is non empty set
n is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
n .last() is Element of the_Vertices_of x
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n . (len n) is set
H is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
H .last() is Element of the_Vertices_of G
len H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H . (len H) is set
y is set
H .vertices() is finite Element of bool (the_Vertices_of G)
H .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(H .vertexSeq())) is finite Element of bool (the_Vertices_of G)
n .vertices() is finite Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty set
n .vertexSeq() is Relation-like NAT -defined the_Vertices_of x -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of x
K426((the_Vertices_of x),(n .vertexSeq())) is finite Element of bool (the_Vertices_of x)
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n . C is set
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H . C is set
the_Vertices_of x is non empty Element of bool (the_Vertices_of G)
y is Element of the_Vertices_of G
n .first() is Element of the_Vertices_of x
n . 1 is set
H .first() is Element of the_Vertices_of G
H . 1 is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x .vertices() is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
x .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(x .vertexSeq())) is finite Element of bool (the_Vertices_of G)
n is Element of the_Vertices_of G
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . H is set
x .first() is Element of the_Vertices_of G
x . 1 is set
x .last() is Element of the_Vertices_of G
x . (len x) is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
G .edgesBetween (the_Vertices_of G) is Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
G .edgesInto (the_Vertices_of G) is Element of bool (the_Edges_of G)
G .edgesOutOf (the_Vertices_of G) is Element of bool (the_Edges_of G)
(G .edgesInto (the_Vertices_of G)) /\ (G .edgesOutOf (the_Vertices_of G)) is Element of bool (the_Edges_of G)
(the_Vertices_of G) \ x is Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
G .edgesBetween ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesInto ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
n is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) is non empty Element of bool (the_Vertices_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) . VertexSelector is set
n .vertices() is finite Element of bool (the_Vertices_of G)
n .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(n .vertexSeq())) is finite Element of bool (the_Vertices_of G)
the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) . EdgeSelector is set
n .edges() is finite Element of bool (the_Edges_of G)
n .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(n .edgeSeq())) is finite Element of bool (the_Edges_of G)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) is non empty set
the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) is set
(the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) is non empty set
n .last() is Element of the_Vertices_of G
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n . (len n) is set
H is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
H .last() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
len H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H . (len H) is set
n .first() is Element of the_Vertices_of G
n . 1 is set
H .first() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
H . 1 is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
(the_Vertices_of G) \ {} is Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
G .edgesBetween ((the_Vertices_of G) \ {}) is Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
G .edgesInto ((the_Vertices_of G) \ {}) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ {}) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ {})) /\ (G .edgesOutOf ((the_Vertices_of G) \ {})) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ {},G .edgesBetween ((the_Vertices_of G) \ {})
the_Vertices_of x is non empty set
x . VertexSelector is set
the_Edges_of x is set
x . EdgeSelector is set
(the_Vertices_of x) \/ (the_Edges_of x) is non empty set
x is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
x .last() is Element of the_Vertices_of x
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . (len x) is set
n is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
n .last() is Element of the_Vertices_of G
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n . (len n) is set
x .first() is Element of the_Vertices_of x
x . 1 is set
n .first() is Element of the_Vertices_of G
n . 1 is set
x .reverse() is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
(x .reverse()) .last() is Element of the_Vertices_of x
len (x .reverse()) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x .reverse()) . (len (x .reverse())) is set
H is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
H .last() is Element of the_Vertices_of G
len H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H . (len H) is set
(x .reverse()) .first() is Element of the_Vertices_of x
(x .reverse()) . 1 is set
H .first() is Element of the_Vertices_of G
H . 1 is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
(the_Vertices_of G) \ x is Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
G .edgesBetween ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of x is non empty set
x . VertexSelector is set
n is Element of the_Vertices_of x
x .reachableFrom n is non empty Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty set
(x .reachableFrom n) /\ x is Element of bool (the_Vertices_of G)
the_Vertices_of x is non empty Element of bool (the_Vertices_of G)
y is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
(the_Vertices_of G) \ x is Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
G .edgesBetween ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of x is non empty set
x . VertexSelector is set
n is Element of the_Vertices_of x
H is Element of the_Vertices_of x
x .reachableFrom n is non empty Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty set
x .reachableFrom H is non empty Element of bool (the_Vertices_of x)
(x .reachableFrom n) /\ (x .reachableFrom H) is Element of bool (the_Vertices_of x)
C is set
the_Edges_of x is set
x . EdgeSelector is set
(the_Vertices_of x) \/ (the_Edges_of x) is non empty set
a is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
b is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
b .reverse() is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
a .append (b .reverse()) is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
(the_Vertices_of G) \ x is Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
G .edgesBetween ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of x is non empty set
x . VertexSelector is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
bool x is non empty set
x is Element of bool x
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
{ b₁ where b₁ is finite (G,S,P) : verum } is set
the finite (G,S,P) is finite (G,S,P)
n is set
bool (the_Vertices_of G) is non empty finite V32() set
H is finite (G,S,P)
H is set
n is non empty finite set
y is finite (G,S,P)
card y is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
[:n,REAL:] is non empty non trivial Relation-like non finite set
bool [:n,REAL:] is non empty non trivial non finite set
H is non empty Relation-like n -defined REAL -valued Function-like finite V39(n) V40(n, REAL ) Element of bool [:n,REAL:]
y is Element of n
H /. y is complex V25() ext-real Element of REAL
H . y is complex V25() ext-real Element of REAL
C is finite (G,S,P)
card C is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
dom H is non empty finite set
bool C is non empty finite V32() set
C is finite Element of bool C
a is Element of n
H . a is complex V25() ext-real Element of REAL
b is finite (G,S,P)
card b is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
H /. b is complex V25() ext-real Element of REAL
H . b is set
card C is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
x is (G,P,S)
bool x is non empty set
n is Element of bool x
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
x is Element of the_Vertices_of G
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
x \ {x} is Element of bool (the_Vertices_of G)
H is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
H .vertices() is finite Element of bool (the_Vertices_of G)
H .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(H .vertexSeq())) is finite Element of bool (the_Vertices_of G)
y is Element of the_Vertices_of G
C is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
(the_Vertices_of G) \ x is Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
G .edgesBetween ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of x is non empty set
x . VertexSelector is set
n is Element of the_Vertices_of x
x .reachableFrom n is non empty Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty set
y is Element of the_Vertices_of G
{y} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
x \ {y} is Element of bool (the_Vertices_of G)
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
C is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
C .vertices() is finite Element of bool (the_Vertices_of G)
C .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(C .vertexSeq())) is finite Element of bool (the_Vertices_of G)
C .find y is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C . (C .find y) is set
C .first() is Element of the_Vertices_of G
C . 1 is set
len C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
a is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C . a is set
C .find (C . a) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- 2 is complex V25() integer ext-real non positive set
(C .find y) + (- 2) is complex V25() integer ext-real set
(len C) + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
3 + (- 2) is complex V25() integer ext-real set
(C .find y) - 2 is complex V25() integer ext-real set
(C .find y) - (2 * 1) is non empty complex V25() integer non even ext-real set
b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
C .cut (1,b) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len (C .cut (1,b)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (C .cut (1,b))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- 1 is complex V25() integer ext-real non positive set
((len (C .cut (1,b))) + 1) + (- 1) is complex V25() integer ext-real set
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(b + 1) + (- 1) is complex V25() integer ext-real set
n1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
(C .cut (1,b)) . n1 is set
C . n1 is set
1 + (- 1) is complex V25() integer ext-real set
n1 + (- 1) is complex V25() integer ext-real set
n1 - 1 is complex V25() integer even ext-real set
bg is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
bg + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(C .cut (1,b)) . (bg + 1) is set
1 + bg is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C . (1 + bg) is set
(C .cut (1,b)) .vertices() is finite Element of bool (the_Vertices_of G)
(C .cut (1,b)) .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),((C .cut (1,b)) .vertexSeq())) is finite Element of bool (the_Vertices_of G)
(b + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x \/ {y} is non empty Element of bool (the_Vertices_of G)
{y} \/ (x \ {y}) is non empty Element of bool (the_Vertices_of G)
C .find (C . n1) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
(C .cut (1,b)) . n1 is set
the_Edges_of x is set
x . EdgeSelector is set
(the_Vertices_of x) \/ (the_Edges_of x) is non empty set
(C .cut (1,b)) . ((2 * 0) + 1) is set
n1 is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
n1 .vertices() is finite Element of bool (the_Vertices_of x)
n1 .vertexSeq() is Relation-like NAT -defined the_Vertices_of x -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of x
K426((the_Vertices_of x),(n1 .vertexSeq())) is finite Element of bool (the_Vertices_of x)
(len C) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len C) + 1) + (- 2) is complex V25() integer ext-real set
(C .cut (1,b)) . b is set
C . b is set
(len C) + (- 1) is complex V25() integer ext-real set
bg is Element of the_Vertices_of G
b + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C . (b + 2) is set
C . (b + 1) is set
C is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
C .vertices() is finite Element of bool (the_Vertices_of G)
C .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(C .vertexSeq())) is finite Element of bool (the_Vertices_of G)
a is Element of the_Vertices_of G
b is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is (G,S,P)
(the_Vertices_of G) \ x is Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
G .edgesBetween ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is Element of bool (the_Edges_of G)
x is (G,P,S)
n is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of n is non empty set
n . VertexSelector is set
H is Element of the_Vertices_of n
n .reachableFrom H is non empty Element of bool (the_Vertices_of n)
bool (the_Vertices_of n) is non empty set
y is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) - 2 is complex V25() integer ext-real set
S .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(S .edgeSeq())) is finite Element of bool (the_Edges_of G)
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . P is set
S . x is set
x is set
S .last() is Element of the_Vertices_of G
S . (len S) is set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len S) - (2 * 1) is non empty complex V25() integer non even ext-real set
S .first() is Element of the_Vertices_of G
S . 1 is set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
H div 2 is complex V25() integer ext-real set
dom (S .edgeSeq()) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
(S .edgeSeq()) . (H div 2) is set
rng (S .edgeSeq()) is finite set
((len S) - 2) + 1 is complex V25() integer ext-real set
S . (((len S) - 2) + 1) is set
S . n is set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (n + 2) is set
S . (n + 1) is set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(2 * 1) div 2 is complex V25() integer ext-real set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .edgeSeq()) . 1 is set
S . (1 + 1) is set
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (1 + 2) is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) - 2 is complex V25() integer ext-real set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . P is set
S . x is set
S .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(S .edgeSeq())) is finite Element of bool (the_Edges_of G)
S .last() is Element of the_Vertices_of G
S . (len S) is set
S .first() is Element of the_Vertices_of G
S . 1 is set
x is set
x is set
dom (S .edgeSeq()) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
(S .edgeSeq()) . n is set
H is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * H is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * H) - 1 is non empty complex V25() integer non even ext-real set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 - 1 is complex V25() integer ext-real set
len (S .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(S .edgeSeq()) . H is set
S . (2 * H) is set
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
C + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . C is set
S . (C + 2) is set
S . (C + 1) is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
S is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
the_Vertices_of S is non empty set
S . VertexSelector is set
the_Edges_of S is set
S . EdgeSelector is set
(the_Vertices_of S) \/ (the_Edges_of S) is non empty set
P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
x is Relation-like NAT -defined (the_Vertices_of S) \/ (the_Edges_of S) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of S
len P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
P .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(P .edgeSeq())) is finite Element of bool (the_Edges_of G)
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . x is set
P . n is set
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . x is set
x .edges() is finite Element of bool (the_Edges_of S)
bool (the_Edges_of S) is non empty set
x .edgeSeq() is Relation-like NAT -defined the_Edges_of S -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of S
K426((the_Edges_of S),(x .edgeSeq())) is finite Element of bool (the_Edges_of S)
x . n is set
H is set
H is set
y is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
the_Vertices_of P is non empty set
P . VertexSelector is set
the_Edges_of P is set
P . EdgeSelector is set
(the_Vertices_of P) \/ (the_Edges_of P) is non empty set
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
x is Relation-like NAT -defined (the_Vertices_of P) \/ (the_Edges_of P) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of P
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x .edges() is finite Element of bool (the_Edges_of P)
bool (the_Edges_of P) is non empty set
x .edgeSeq() is Relation-like NAT -defined the_Edges_of P -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of P
K426((the_Edges_of P),(x .edgeSeq())) is finite Element of bool (the_Edges_of P)
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . n is set
x . H is set
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . n is set
x .edges() is finite Element of bool (the_Edges_of G)
x .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(x .edgeSeq())) is finite Element of bool (the_Edges_of G)
x . H is set
y is set
y is set
C is set
the_Vertices_of P is non empty Element of bool (the_Vertices_of G)
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x .edges() is finite Element of bool (the_Edges_of G)
x .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(x .edgeSeq())) is finite Element of bool (the_Edges_of G)
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . n is set
x . H is set
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . n is set
x .edges() is finite Element of bool (the_Edges_of P)
bool (the_Edges_of P) is non empty set
x .edgeSeq() is Relation-like NAT -defined the_Edges_of P -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of P
K426((the_Edges_of P),(x .edgeSeq())) is finite Element of bool (the_Edges_of P)
x . H is set
y is set
y is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (S .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S . 1 is set
S . 5 is set
S . 3 is set
S . 7 is set
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 4 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 4) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) - 2 is complex V25() integer ext-real set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . P is set
S . x is set
x is set
x is set
9 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
9 - 2 is complex V25() integer ext-real set
2 * 3 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 3) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
7 - 2 is complex V25() integer ext-real set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n - 2 is complex V25() integer ext-real set
2 * 2 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
5 - 2 is complex V25() integer ext-real set
S .first() is Element of the_Vertices_of G
S .last() is Element of the_Vertices_of G
S . (len S) is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
3 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
5 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
7 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) - 2 is complex V25() integer ext-real set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . P is set
S . x is set
x is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S .first() is Element of the_Vertices_of G
S . 1 is set
S .last() is Element of the_Vertices_of G
S . (len S) is set
(len S) - 2 is complex V25() integer ext-real set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . P is set
S . x is set
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
3 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .reverse() is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(S .edgeSeq())) is finite Element of bool (the_Edges_of G)
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . x is set
S . x is set
n is set
n is set
(len S) - x is complex V25() integer even ext-real set
((len S) - x) + 1 is non empty complex V25() integer non even ext-real set
(len S) - x is complex V25() integer even ext-real set
((len S) - x) + 1 is non empty complex V25() integer non even ext-real set
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
a is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
b + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
(S .reverse()) . b is set
len (S .reverse()) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .reverse()) . a is set
(S .reverse()) .edges() is finite Element of bool (the_Edges_of G)
(S .reverse()) .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),((S .reverse()) .edgeSeq())) is finite Element of bool (the_Edges_of G)
v is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .reverse() is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .reverse()) .reverse() is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . P is set
S . x is set
P + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is set
S .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(S .edgeSeq())) is finite Element of bool (the_Edges_of G)
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (n + 1) is set
S . n is set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (n + 2) is set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
n + (2 * 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is set
P + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (P + 1) is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S .cut (P,x) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
len (S .cut (P,x)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .cut (P,x)) .edges() is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
(S .cut (P,x)) .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),((S .cut (P,x)) .edgeSeq())) is finite Element of bool (the_Edges_of G)
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S .cut (P,x)) . n is set
(S .cut (P,x)) . H is set
y is set
y is set
P + n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real positive non negative set
(P + n) - 1 is non empty complex V25() integer non even ext-real set
P + H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real positive non negative set
(P + H) - 1 is non empty complex V25() integer non even ext-real set
dom (S .cut (P,x)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
S . ((P + H) - 1) is set
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
n + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
a is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
v is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H + P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real positive non negative set
(n + 2) + P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(P + n) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((P + n) + 2) - 1 is non empty complex V25() integer non even ext-real set
n1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . n1 is set
S . b is set
(S .cut (P,x)) .first() is Element of the_Vertices_of G
(S .cut (P,x)) . 1 is set
(S .cut (P,x)) .last() is Element of the_Vertices_of G
(S .cut (P,x)) . (len (S .cut (P,x))) is set
S . x is set
S . P is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is non empty Element of bool (the_Vertices_of G)
G .edgesBetween S is Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
the_Vertices_of P is non empty set
P . VertexSelector is set
the_Edges_of P is set
P . EdgeSelector is set
(the_Vertices_of P) \/ (the_Edges_of P) is non empty set
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
x is Relation-like NAT -defined (the_Vertices_of P) \/ (the_Edges_of P) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of P
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x .edges() is finite Element of bool (the_Edges_of P)
bool (the_Edges_of P) is non empty set
x .edgeSeq() is Relation-like NAT -defined the_Edges_of P -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of P
K426((the_Edges_of P),(x .edgeSeq())) is finite Element of bool (the_Edges_of P)
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . n is set
x . H is set
y is set
y is set
x .vertices() is finite Element of bool (the_Vertices_of P)
bool (the_Vertices_of P) is non empty set
x .vertexSeq() is Relation-like NAT -defined the_Vertices_of P -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of P
K426((the_Vertices_of P),(x .vertexSeq())) is finite Element of bool (the_Vertices_of P)
the_Vertices_of P is non empty Element of bool (the_Vertices_of G)
n + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x .edges() is finite Element of bool (the_Edges_of G)
x .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(x .edgeSeq())) is finite Element of bool (the_Edges_of G)
x . n is set
x . H is set
C is set
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . n is set
x . H is set
y is set
y is set
x . n is set
x . H is set
C is set
n + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite [Graph-like] set
S is Relation-like NAT -defined Function-like finite [Graph-like] set
the_Vertices_of S is non empty set
S . VertexSelector is set
the_Edges_of S is set
S . EdgeSelector is set
(the_Vertices_of S) \/ (the_Edges_of S) is non empty set
P is Relation-like NAT -defined (the_Vertices_of S) \/ (the_Edges_of S) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of S
P .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
P .edgeSeq() is Relation-like NAT -defined the_Edges_of S -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of S
len (P .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
x .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (x .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (x .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (x .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * (P .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (P .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
card (the_Vertices_of G) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of omega
2 * 3 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 3) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 2 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Edges_of G is finite set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty finite set
n is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
n .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
n .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (n .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n . P is set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n . x is set
S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n . S is set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n . x is set
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 4 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (n .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
8 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(2 * (n .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
9 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
a is set
{(n . x),(n . x),(n . P),(n . S)} is non empty finite set
card {(n . x),(n . x),(n . P),(n . S)} is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of omega
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
S .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (S .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set is Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set . VertexSelector is set
S is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set
{S} is non empty trivial finite 1 -element Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set )
bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set ) is non empty finite V32() set
2 * 2 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set is finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set . EdgeSelector is set
(the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set ) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set ) is non empty finite set
x is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set ) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set ) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set
x .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x .edgeSeq() is Relation-like NAT -defined the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set
len (x .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (x .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * 3 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (x .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x . x is set
x .vertices() is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set )
x .vertexSeq() is Relation-like NAT -defined the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set
K426((the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set ),(x .vertexSeq())) is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set )
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x . P is set
P is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set ) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set ) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set
P .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
P .edgeSeq() is Relation-like NAT -defined the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of the Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () set
len (P .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
{0,1} is non empty finite V32() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
{0} is non empty trivial functional finite V32() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
0 .--> 0 is Relation-like NAT -defined {0} -defined NAT -valued Function-like one-to-one finite Function-yielding V100() set
{0} is non empty trivial functional finite V32() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() set
{0} --> 0 is non empty Relation-like {0} -defined NAT -valued {0} -valued Function-like constant finite V39({0}) V40({0},{0}) Function-yielding V100() Element of bool [:{0},{0}:]
[:{0},{0}:] is non empty Relation-like finite set
bool [:{0},{0}:] is non empty finite V32() set
0 .--> 1 is Relation-like NAT -defined {0} -defined NAT -valued Function-like one-to-one finite set
{0} --> 1 is non empty Relation-like non-empty {0} -defined NAT -valued {1} -valued Function-like constant finite V39({0}) V40({0},{1}) Element of bool [:{0},{1}:]
{1} is non empty trivial finite V32() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() set
[:{0},{1}:] is non empty Relation-like finite set
bool [:{0},{1}:] is non empty finite V32() set
dom (0 .--> 1) is finite set
x is set
(0 .--> 1) . x is set
x is set
(0 .--> 0) . x is Relation-like Function-like set
[:{0},{0,1}:] is non empty Relation-like finite set
bool [:{0},{0,1}:] is non empty finite V32() set
dom (0 .--> 0) is finite set
n is non empty Relation-like {0} -defined {0,1} -valued Function-like finite V39({0}) V40({0},{0,1}) Element of bool [:{0},{0,1}:]
x is non empty Relation-like {0} -defined {0,1} -valued Function-like finite V39({0}) V40({0},{0,1}) Element of bool [:{0},{0,1}:]
createGraph ({0,1},{0},n,x) is Relation-like NAT -defined Function-like finite [Graph-like] finite set
<*{0,1},{0},n,x*> is set
the_Edges_of (createGraph ({0,1},{0},n,x)) is finite set
(createGraph ({0,1},{0},n,x)) . EdgeSelector is set
the_Vertices_of (createGraph ({0,1},{0},n,x)) is non empty finite set
(createGraph ({0,1},{0},n,x)) . VertexSelector is set
the_Source_of (createGraph ({0,1},{0},n,x)) is Relation-like the_Edges_of (createGraph ({0,1},{0},n,x)) -defined the_Vertices_of (createGraph ({0,1},{0},n,x)) -valued Function-like finite V39( the_Edges_of (createGraph ({0,1},{0},n,x))) V40( the_Edges_of (createGraph ({0,1},{0},n,x)), the_Vertices_of (createGraph ({0,1},{0},n,x))) Element of bool [:(the_Edges_of (createGraph ({0,1},{0},n,x))),(the_Vertices_of (createGraph ({0,1},{0},n,x))):]
[:(the_Edges_of (createGraph ({0,1},{0},n,x))),(the_Vertices_of (createGraph ({0,1},{0},n,x))):] is Relation-like finite set
bool [:(the_Edges_of (createGraph ({0,1},{0},n,x))),(the_Vertices_of (createGraph ({0,1},{0},n,x))):] is non empty finite V32() set
(createGraph ({0,1},{0},n,x)) . SourceSelector is set
(the_Source_of (createGraph ({0,1},{0},n,x))) . 0 is set
card (the_Vertices_of (createGraph ({0,1},{0},n,x))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of omega
y is set
{y} is non empty trivial finite 1 -element set
the_Target_of (createGraph ({0,1},{0},n,x)) is Relation-like the_Edges_of (createGraph ({0,1},{0},n,x)) -defined the_Vertices_of (createGraph ({0,1},{0},n,x)) -valued Function-like finite V39( the_Edges_of (createGraph ({0,1},{0},n,x))) V40( the_Edges_of (createGraph ({0,1},{0},n,x)), the_Vertices_of (createGraph ({0,1},{0},n,x))) Element of bool [:(the_Edges_of (createGraph ({0,1},{0},n,x))),(the_Vertices_of (createGraph ({0,1},{0},n,x))):]
(createGraph ({0,1},{0},n,x)) . TargetSelector is set
(the_Target_of (createGraph ({0,1},{0},n,x))) . 0 is set
y is set
C is set
C is Element of the_Vertices_of (createGraph ({0,1},{0},n,x))
C is set
a is set
y is set
C is set
G is Relation-like NAT -defined Function-like finite [Graph-like] set
2 * 3 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 3) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
x .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (x .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 4 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (x .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
8 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(2 * (x .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
9 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x . P is set
S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x . S is set
x is Element of the_Vertices_of G
n is Element of the_Vertices_of G
P + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H is set
G is Relation-like NAT -defined Function-like finite [Graph-like] () set
S is set
G .edgesBetween S is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto S is Element of bool (the_Edges_of G)
G .edgesOutOf S is Element of bool (the_Edges_of G)
(G .edgesInto S) /\ (G .edgesOutOf S) is Element of bool (the_Edges_of G)
P is Relation-like NAT -defined Function-like finite [Graph-like] inducedSubgraph of G,S,G .edgesBetween S
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
the_Vertices_of P is non empty Element of bool (the_Vertices_of G)
P . VertexSelector is set
the_Vertices_of P is non empty set
the_Edges_of P is set
P . EdgeSelector is set
(the_Vertices_of P) \/ (the_Edges_of P) is non empty set
x is Relation-like NAT -defined (the_Vertices_of P) \/ (the_Edges_of P) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of P
x .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x .edgeSeq() is Relation-like NAT -defined the_Edges_of P -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of P
len (x .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
n is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
n .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
n .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (n .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
len n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len n) - 2 is complex V25() integer ext-real set
y is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
H + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n . H is set
n . y is set
C is set
C is set
H + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . y is set
the_Vertices_of G is non empty set
G . VertexSelector is set
bool (the_Vertices_of G) is non empty set
G is Relation-like NAT -defined Function-like finite [Graph-like] () set
the_Vertices_of G is non empty set
G . VertexSelector is set
the_Edges_of G is set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
S is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
S .vertices() is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
S .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(S .vertexSeq())) is finite Element of bool (the_Vertices_of G)
S .last() is Element of the_Vertices_of G
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S . (len S) is set
(len S) - 2 is complex V25() integer ext-real set
S . ((len S) - 2) is set
P is set
x is set
S .addEdge x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(S .last()) .adj x is Element of the_Vertices_of G
G .walkOf ((S .last()),x,((S .last()) .adj x)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
S .append (G .walkOf ((S .last()),x,((S .last()) .adj x))) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
x is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
S . n is set
x . n is set
x .last() is Element of the_Vertices_of G
x . (len x) is set
n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
2 * n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
H + (2 * n) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . H is set
H + 4 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(H + 4) - 4 is complex V25() integer ext-real set
((len S) + 2) - 4 is complex V25() integer ext-real set
y is set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len S) - (2 * 1) is non empty complex V25() integer non even ext-real set
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x .cut (C,C) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
(x .cut (C,C)) .last() is Element of the_Vertices_of G
len (x .cut (C,C)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x .cut (C,C)) . (len (x .cut (C,C))) is set
(len (x .cut (C,C))) + H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) + 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (x .cut (C,C))) - 1 is complex V25() integer even ext-real set
b is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
((len (x .cut (C,C))) - 1) + H is non empty complex V25() integer non even ext-real set
H + b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
((len S) + 2) - 2 is complex V25() integer ext-real set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x .cut (C,C)) . (b + 1) is set
x . (H + b) is set
S . (H + b) is set
(x .cut (C,C)) .addEdge y is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
((x .cut (C,C)) .last()) .adj y is Element of the_Vertices_of G
G .walkOf (((x .cut (C,C)) .last()),y,(((x .cut (C,C)) .last()) .adj y)) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
(x .cut (C,C)) .append (G .walkOf (((x .cut (C,C)) .last()),y,(((x .cut (C,C)) .last()) .adj y))) is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
(x .cut (C,C)) .first() is Element of the_Vertices_of G
(x .cut (C,C)) . 1 is set
v is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
dom (x .cut (C,C)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
((x .cut (C,C)) .addEdge y) . v is set
(x .cut (C,C)) . v is set
(len S) + 3 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) + 3) - H is complex V25() integer ext-real set
((len S) + 3) - ((len S) - 2) is complex V25() integer ext-real set
(len (x .cut (C,C))) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
5 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len ((x .cut (C,C)) .addEdge y) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((x .cut (C,C)) .addEdge y) .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
((x .cut (C,C)) .addEdge y) .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (((x .cut (C,C)) .addEdge y) .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (((x .cut (C,C)) .addEdge y) .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * 3 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * (((x .cut (C,C)) .addEdge y) .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(2 * 3) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S .vertexAt H is Element of the_Vertices_of G
S . H is set
(len ((x .cut (C,C)) .addEdge y)) - 2 is complex V25() integer ext-real set
n1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
v is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
v + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((x .cut (C,C)) .addEdge y) . v is set
((x .cut (C,C)) .addEdge y) . n1 is set
bg is set
bg is set
1 - 1 is complex V25() integer ext-real set
v - 1 is complex V25() integer even ext-real set
a is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
((len (x .cut (C,C))) + 2) - 2 is complex V25() integer ext-real set
(v + 2) - 2 is complex V25() integer ext-real set
aa is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
aa + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x .cut (C,C)) . (aa + 1) is set
H + aa is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
S . (H + aa) is set
((x .cut (C,C)) .addEdge y) .last() is Element of the_Vertices_of G
((x .cut (C,C)) .addEdge y) . (len ((x .cut (C,C)) .addEdge y)) is set
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(2 + 1) - 1 is complex V25() integer ext-real set
H + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (x .cut (C,C))) + (2 * 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x .cut (C,C)) . n1 is set
n1 - 1 is complex V25() integer even ext-real set
bb is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(v + 2) - 1 is complex V25() integer even ext-real set
e is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
H + e is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
aa + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H + (aa + 2) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(H + aa) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x .cut (C,C)) . v is set
x . (H + aa) is set
x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
2 * x is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(H + aa) + (2 * x) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H + (2 * x) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(H + (2 * x)) + aa is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x .cut (C,C)) . (e + 1) is set
S . (H + e) is set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len S) - (2 * 1) is non empty complex V25() integer non even ext-real set
n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
2 * n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
H + (2 * n) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . H is set
y is set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
((len S) - 2) + 4 is complex V25() integer ext-real set
H is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
2 * H is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
n + (2 * H) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + ((len S) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n + 4 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . n is set
y is set
(len x) - 2 is complex V25() integer ext-real set
H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
x . n is set
x . H is set
((len S) + 2) - 2 is complex V25() integer ext-real set
(n + 2) - 2 is complex V25() integer ext-real set
y is set
S . n is set
S . H is set
y is set
x .first() is Element of the_Vertices_of G
x . 1 is set
S .first() is Element of the_Vertices_of G
S . 1 is set
G is Relation-like NAT -defined Function-like finite [Graph-like] () set
the_Vertices_of G is non empty set
G . VertexSelector is set
P is Element of the_Vertices_of G
x is Element of the_Vertices_of G
x is (G,P,x)
G .edgesBetween x is Element of bool (the_Edges_of G)
the_Edges_of G is set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty set
G .edgesInto x is Element of bool (the_Edges_of G)
G .edgesOutOf x is Element of bool (the_Edges_of G)
(G .edgesInto x) /\ (G .edgesOutOf x) is Element of bool (the_Edges_of G)
(the_Vertices_of G) \ x is Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty set
G .edgesBetween ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesInto ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) is Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) is non empty set
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) . VertexSelector is set
H is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H is non empty Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) is non empty set
y is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y is non empty Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H) /\ ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y) is Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y) is Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) is set
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) . EdgeSelector is set
bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) is non empty set
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesInto ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y) is Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesOutOf ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y) is Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesInto ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y)) /\ ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesOutOf ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y)) is Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
the Relation-like NAT -defined Function-like finite [Graph-like] connected Component-like () inducedSubgraph of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x), the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y, the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y) is Relation-like NAT -defined Function-like finite [Graph-like] connected Component-like () inducedSubgraph of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x), the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y, the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H) is Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesInto ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H) is Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesOutOf ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H) is Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesInto ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H)) /\ ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesOutOf ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H)) is Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
the Relation-like NAT -defined Function-like finite [Graph-like] connected Component-like () inducedSubgraph of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x), the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H, the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H) is Relation-like NAT -defined Function-like finite [Graph-like] connected Component-like () inducedSubgraph of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x), the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H, the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] connected Component-like () inducedSubgraph of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x), the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H, the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H) is non empty Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
the Relation-like NAT -defined Function-like finite [Graph-like] connected Component-like () inducedSubgraph of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x), the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H, the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H) . VertexSelector is set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] connected Component-like () inducedSubgraph of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x), the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y, the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y) is non empty Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x))
the Relation-like NAT -defined Function-like finite [Graph-like] connected Component-like () inducedSubgraph of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x), the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y, the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y) . VertexSelector is set
( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom y) /\ x is Element of bool (the_Vertices_of G)
( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .reachableFrom H) /\ x is Element of bool (the_Vertices_of G)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) is non empty Element of bool (the_Vertices_of G)
bg is Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,x,G .edgesBetween x
the_Vertices_of bg is non empty set
bg . VertexSelector is set
a is Element of the_Vertices_of bg
aa is Element of the_Vertices_of bg
the_Vertices_of bg is non empty Element of bool (the_Vertices_of G)
v is non empty Element of bool (the_Vertices_of G)
bb is Element of the_Vertices_of G
n1 is non empty Element of bool (the_Vertices_of G)
{bb} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
n1 \/ {bb} is non empty Element of bool (the_Vertices_of G)
v \/ {bb} is non empty Element of bool (the_Vertices_of G)
G .edgesBetween (n1 \/ {bb}) is Element of bool (the_Edges_of G)
G .edgesInto (n1 \/ {bb}) is Element of bool (the_Edges_of G)
G .edgesOutOf (n1 \/ {bb}) is Element of bool (the_Edges_of G)
(G .edgesInto (n1 \/ {bb})) /\ (G .edgesOutOf (n1 \/ {bb})) is Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,n1 \/ {bb},G .edgesBetween (n1 \/ {bb}) is Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,n1 \/ {bb},G .edgesBetween (n1 \/ {bb})
G .edgesBetween (v \/ {bb}) is Element of bool (the_Edges_of G)
G .edgesInto (v \/ {bb}) is Element of bool (the_Edges_of G)
G .edgesOutOf (v \/ {bb}) is Element of bool (the_Edges_of G)
(G .edgesInto (v \/ {bb})) /\ (G .edgesOutOf (v \/ {bb})) is Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,v \/ {bb},G .edgesBetween (v \/ {bb}) is Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,v \/ {bb},G .edgesBetween (v \/ {bb})
e is Element of the_Vertices_of G
{bb,e} is non empty finite Element of bool (the_Vertices_of G)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,n1 \/ {bb},G .edgesBetween (n1 \/ {bb}) is non empty Element of bool (the_Vertices_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,n1 \/ {bb},G .edgesBetween (n1 \/ {bb}) . VertexSelector is set
j is set
j is Element of the_Vertices_of G
{a} is non empty trivial finite 1 -element Element of bool (the_Vertices_of bg)
bool (the_Vertices_of bg) is non empty set
n1 \/ {a} is non empty set
{aa} is non empty trivial finite 1 -element Element of bool (the_Vertices_of bg)
(n1 \/ {a}) \/ {aa} is non empty set
G .edgesBetween ((n1 \/ {a}) \/ {aa}) is Element of bool (the_Edges_of G)
G .edgesInto ((n1 \/ {a}) \/ {aa}) is Element of bool (the_Edges_of G)
G .edgesOutOf ((n1 \/ {a}) \/ {aa}) is Element of bool (the_Edges_of G)
(G .edgesInto ((n1 \/ {a}) \/ {aa})) /\ (G .edgesOutOf ((n1 \/ {a}) \/ {aa})) is Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}) is Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
v \/ {a} is non empty set
(v \/ {a}) \/ {aa} is non empty set
G .edgesBetween ((v \/ {a}) \/ {aa}) is Element of bool (the_Edges_of G)
G .edgesInto ((v \/ {a}) \/ {aa}) is Element of bool (the_Edges_of G)
G .edgesOutOf ((v \/ {a}) \/ {aa}) is Element of bool (the_Edges_of G)
(G .edgesInto ((v \/ {a}) \/ {aa})) /\ (G .edgesOutOf ((v \/ {a}) \/ {aa})) is Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}) is Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,v \/ {bb},G .edgesBetween (v \/ {bb}) is non empty Element of bool (the_Vertices_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,v \/ {bb},G .edgesBetween (v \/ {bb}) . VertexSelector is set
G .edgesBetween v is Element of bool (the_Edges_of G)
G .edgesInto v is Element of bool (the_Edges_of G)
G .edgesOutOf v is Element of bool (the_Edges_of G)
(G .edgesInto v) /\ (G .edgesOutOf v) is Element of bool (the_Edges_of G)
x is set
G .edgesBetween n1 is Element of bool (the_Edges_of G)
G .edgesInto n1 is Element of bool (the_Edges_of G)
G .edgesOutOf n1 is Element of bool (the_Edges_of G)
(G .edgesInto n1) /\ (G .edgesOutOf n1) is Element of bool (the_Edges_of G)
v \/ {bb,e} is non empty Element of bool (the_Vertices_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,n1,G .edgesBetween n1
the_Vertices_of x is non empty Element of bool (the_Vertices_of G)
x . VertexSelector is set
(G,(the_Vertices_of x)) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in the_Vertices_of x & ex b₂ being Element of the_Vertices_of G st
( b₂ in the_Vertices_of x & (G,b₁,b₂) ) ) } is set
x is Element of the_Vertices_of G
n1 \/ {bb,e} is non empty Element of bool (the_Vertices_of G)
x is Element of the_Vertices_of G
{a} \/ {aa} is non empty finite Element of bool (the_Vertices_of bg)
n1 \/ ({a} \/ {aa}) is non empty set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}) is non empty Element of bool (the_Vertices_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}) . VertexSelector is set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}) is non empty set
(G,(the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,n1 \/ {bb},G .edgesBetween (n1 \/ {bb}))) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,n1 \/ {bb},G .edgesBetween (n1 \/ {bb}) & ex b₂ being Element of the_Vertices_of G st
( b₂ in the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,n1 \/ {bb},G .edgesBetween (n1 \/ {bb}) & (G,b₁,b₂) ) ) } is set
the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}) is set
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}) . EdgeSelector is set
(the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})) is non empty set
k is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
k is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
Q1 is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
gds is Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,v,G .edgesBetween v
the_Vertices_of gds is non empty Element of bool (the_Vertices_of G)
gds . VertexSelector is set
(G,(the_Vertices_of gds)) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in the_Vertices_of gds & ex b₂ being Element of the_Vertices_of G st
( b₂ in the_Vertices_of gds & (G,b₁,b₂) ) ) } is set
Q is Element of the_Vertices_of G
v \/ ({a} \/ {aa}) is non empty set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}) is non empty Element of bool (the_Vertices_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}) . VertexSelector is set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}) is non empty set
(G,(the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,v \/ {bb},G .edgesBetween (v \/ {bb}))) is Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,v \/ {bb},G .edgesBetween (v \/ {bb}) & ex b₂ being Element of the_Vertices_of G st
( b₂ in the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,v \/ {bb},G .edgesBetween (v \/ {bb}) & (G,b₁,b₂) ) ) } is set
the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}) is set
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}) . EdgeSelector is set
(the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})) is non empty set
Q is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
cc is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
dd is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
(v \/ {bb,e}) /\ (n1 \/ {bb,e}) is Element of bool (the_Vertices_of G)
v /\ n1 is Element of bool (the_Vertices_of G)
(v /\ n1) \/ {bb,e} is non empty Element of bool (the_Vertices_of G)
dd .last() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
len dd is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
dd . (len dd) is set
dd .first() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
dd . 1 is set
Q1 .last() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
len Q1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
Q1 . (len Q1) is set
Q1 .first() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
Q1 . 1 is set
xs is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
ys is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
(the_Vertices_of G) \/ (the_Edges_of G) is non empty set
xs .first() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
xs . 1 is set
R is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
R . 1 is set
R .vertices() is finite Element of bool (the_Vertices_of G)
R .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(R .vertexSeq())) is finite Element of bool (the_Vertices_of G)
xs .vertices() is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}))
bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})) is non empty set
xs .vertexSeq() is Relation-like NAT -defined the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}) -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
K426((the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})),(xs .vertexSeq())) is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}))
ej is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
ej .vertices() is finite Element of bool (the_Vertices_of G)
ej .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(ej .vertexSeq())) is finite Element of bool (the_Vertices_of G)
ys .vertices() is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}))
bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})) is non empty set
ys .vertexSeq() is Relation-like NAT -defined the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}) -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
K426((the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})),(ys .vertexSeq())) is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}))
(ej .vertices()) /\ (R .vertices()) is finite Element of bool (the_Vertices_of G)
ej .append R is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
ys .last() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
len ys is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
ys . (len ys) is set
len ej is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
ej . (len ej) is set
xs .last() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
len xs is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
xs . (len xs) is set
len R is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (len R) is set
R .last() is Element of the_Vertices_of G
R .first() is Element of the_Vertices_of G
R .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
R .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (R .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
ys .first() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
ys . 1 is set
ej . 1 is set
ej .edges() is finite Element of bool (the_Edges_of G)
ej .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(ej .edgeSeq())) is finite Element of bool (the_Edges_of G)
R .edges() is finite Element of bool (the_Edges_of G)
K426((the_Edges_of G),(R .edgeSeq())) is finite Element of bool (the_Edges_of G)
(ej .edges()) /\ (R .edges()) is finite Element of bool (the_Edges_of G)
m is set
ys .edges() is finite Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}))
bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})) is non empty set
ys .edgeSeq() is Relation-like NAT -defined the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}) -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
K426((the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})),(ys .edgeSeq())) is finite Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa}))
m1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
ej . m1 is set
m1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
ej . (m1 + 1) is set
e is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(v \/ {a}) \/ {aa},G .edgesBetween ((v \/ {a}) \/ {aa})
ej . (m1 + 2) is set
xs .edges() is finite Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}))
bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})) is non empty set
xs .edgeSeq() is Relation-like NAT -defined the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}) -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
K426((the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})),(xs .edgeSeq())) is finite Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa}))
m1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n1 is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
R . m1 is set
m1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (m1 + 1) is set
n1 is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(n1 \/ {a}) \/ {aa},G .edgesBetween ((n1 \/ {a}) \/ {aa})
R . (m1 + 2) is set
ej .first() is Element of the_Vertices_of G
ej .last() is Element of the_Vertices_of G
len (ej .append R) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (ej .append R)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len ej) + (len R) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
ej .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
len (ej .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(ej .append R) .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(ej .append R) .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len ((ej .append R) .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(ej .length()) + (R .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m is set
(ej .append R) .edges() is finite Element of bool (the_Edges_of G)
K426((the_Edges_of G),((ej .append R) .edgeSeq())) is finite Element of bool (the_Edges_of G)
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
m is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
m + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(ej .append R) . m is set
(ej .append R) . n is set
e is set
e is set
dom ej is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
ej . n is set
ej . m is set
m1 is set
ej . m is set
dom ej is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
dom (ej .append R) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
m1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len ej) + m1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(ej .append R) . ((len ej) + m1) is set
m1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (m1 + 1) is set
n1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
n1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . n1 is set
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
ej . (1 + 1) is set
(ej .append R) . (1 + 1) is set
(len ej) - 1 is complex V25() integer even ext-real set
((len ej) - 1) + (len R) is non empty complex V25() integer non even ext-real set
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(m + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
ej . (m + 1) is set
(ej .append R) . (m + 1) is set
(len ej) - 1 is complex V25() integer even ext-real set
((len ej) - 1) + (len R) is non empty complex V25() integer non even ext-real set
m + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m1 is set
the_Vertices_of x is non empty set
xs . n1 is set
the_Vertices_of gds is non empty set
ys . m is set
m11 is Element of the_Vertices_of gds
m1 is Element of the_Vertices_of x
the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .walkOf (m11,e,m1) is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
(the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) is non empty set
the_Edges_of gds is set
gds . EdgeSelector is set
(the_Vertices_of gds) \/ (the_Edges_of gds) is non empty set
aa is Element of the_Vertices_of gds
WA is Relation-like NAT -defined (the_Vertices_of gds) \/ (the_Edges_of gds) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of gds
the_Edges_of x is set
x . EdgeSelector is set
(the_Vertices_of x) \/ (the_Edges_of x) is non empty set
bb is Element of the_Vertices_of x
WB is Relation-like NAT -defined (the_Vertices_of x) \/ (the_Edges_of x) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of x
WA is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
WB is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
WBs is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
WAs is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
WAs .append ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .walkOf (m11,e,m1)) is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
(WAs .append ( the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x) .walkOf (m11,e,m1))) .append WBs is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
n1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(n1 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len R) - 1 is complex V25() integer even ext-real set
((len R) - 1) + (len ej) is non empty complex V25() integer non even ext-real set
(len ej) + n1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (n1 + 1) is set
(ej .append R) . ((len ej) + n1) is set
m1 is set
m1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
(len ej) + m1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
dom ej is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
dom (ej .append R) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
n1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len ej) + n1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(ej .append R) . ((len ej) + n1) is set
n1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (n1 + 1) is set
(ej .append R) . ((len ej) + m1) is set
m1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (m1 + 1) is set
m1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
m1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
n1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n11 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m11 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m11 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m11 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len R) - 1 is complex V25() integer even ext-real set
((len R) - 1) + (len ej) is non empty complex V25() integer non even ext-real set
(len ej) + m11 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . m11 is set
R . n11 is set
R . (m11 + 1) is set
(ej .append R) . ((len ej) + m11) is set
bb is set
2 * 2 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of P is non empty finite set
P . VertexSelector is set
the_Edges_of P is finite set
P . EdgeSelector is set
(the_Vertices_of P) \/ (the_Edges_of P) is non empty finite set
x is Relation-like NAT -defined (the_Vertices_of P) \/ (the_Edges_of P) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of P
x .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x .edgeSeq() is Relation-like NAT -defined the_Edges_of P -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of P
len (x .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 * 4 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
2 * (x .length()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
8 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(2 * (x .length())) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
9 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x . S is set
G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x . G is set
S + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- 2 is complex V25() integer ext-real non positive set
9 + (- 2) is complex V25() integer ext-real set
(len x) + (- 2) is complex V25() integer ext-real set
(len x) - 2 is complex V25() integer ext-real set
x is set
x .cut (G,(len x)) is Relation-like NAT -defined (the_Vertices_of P) \/ (the_Edges_of P) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of P
n is Element of the_Vertices_of P
x is Element of the_Vertices_of P
y is finite (P,n,x)
P .edgesBetween y is finite Element of bool (the_Edges_of P)
bool (the_Edges_of P) is non empty finite V32() set
P .edgesInto y is finite Element of bool (the_Edges_of P)
P .edgesOutOf y is finite Element of bool (the_Edges_of P)
(P .edgesInto y) /\ (P .edgesOutOf y) is finite Element of bool (the_Edges_of P)
the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of P,y,P .edgesBetween y is Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of P,y,P .edgesBetween y
x . (len x) is set
len (x .cut (G,(len x))) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
(x .cut (G,(len x))) . C is set
- 1 is complex V25() integer ext-real non positive set
C + (- 1) is complex V25() integer ext-real set
1 + (- 1) is complex V25() integer ext-real set
C - 1 is complex V25() integer even ext-real set
a is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len (x .cut (G,(len x)))) + (- 1) is complex V25() integer ext-real set
(len (x .cut (G,(len x)))) + 5 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- 5 is complex V25() integer ext-real non positive set
((len (x .cut (G,(len x)))) + 5) + (- 5) is complex V25() integer ext-real set
(len x) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len x) + 1) + (- 5) is complex V25() integer ext-real set
(len x) - 5 is complex V25() integer ext-real set
((len x) - 5) + G is complex V25() integer ext-real set
b is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even ext-real non negative set
b + G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
C + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x .cut (G,(len x))) . (b + 1) is set
G + b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
x . (G + b) is set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of P,y,P .edgesBetween y is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of P,y,P .edgesBetween y . VertexSelector is set
x .cut (S,G) is Relation-like NAT -defined (the_Vertices_of P) \/ (the_Edges_of P) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of P
len (x .cut (S,G)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (x .cut (S,G))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len (x .cut (S,G))) + 1) + (- 1) is complex V25() integer ext-real set
5 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(5 + 1) + (- 1) is complex V25() integer ext-real set
bg is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
(x .cut (S,G)) . bg is set
5 - 2 is complex V25() integer ext-real set
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x .cut (S,G)) . (2 + 1) is set
x . (1 + 2) is set
x . 3 is set
bg + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(bg + 2) + b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(bg + 2) + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
a is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of P,y,P .edgesBetween y
v is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of P,y,P .edgesBetween y
aa is set
x . bg is set
x is set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite () set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is finite (G,S,P)
(the_Vertices_of G) \ x is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty finite V32() set
G .edgesBetween ((the_Vertices_of G) \ x) is finite Element of bool (the_Edges_of G)
the_Edges_of G is finite set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty finite V32() set
G .edgesInto ((the_Vertices_of G) \ x) is finite Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is finite Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is finite Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of x is non empty finite set
x . VertexSelector is set
n is Element of the_Vertices_of x
x .reachableFrom n is non empty finite Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty finite V32() set
H is Element of the_Vertices_of G
H is non empty finite Element of bool (the_Vertices_of G)
{ H₁(b₁) where b₁ is Element of the_Vertices_of G : b₁ in x .reachableFrom n } is set
C is set
a is Element of the_Vertices_of G
{a} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
(G,{a}) is finite Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {a} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {a} & (G,b₁,b₂) ) ) } is set
(G,{a}) /\ H is finite Element of bool (the_Vertices_of G)
card ((G,{a}) /\ H) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
{S} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
(G,{S}) is finite Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {S} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {S} & (G,b₁,b₂) ) ) } is set
(G,{S}) /\ H is finite Element of bool (the_Vertices_of G)
card ((G,{S}) /\ H) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
x .edgesBetween (x .reachableFrom n) is finite Element of bool (the_Edges_of x)
the_Edges_of x is finite set
x . EdgeSelector is set
bool (the_Edges_of x) is non empty finite V32() set
x .edgesInto (x .reachableFrom n) is finite Element of bool (the_Edges_of x)
x .edgesOutOf (x .reachableFrom n) is finite Element of bool (the_Edges_of x)
(x .edgesInto (x .reachableFrom n)) /\ (x .edgesOutOf (x .reachableFrom n)) is finite Element of bool (the_Edges_of x)
the Relation-like NAT -defined Function-like finite [Graph-like] finite connected Component-like () inducedSubgraph of x,x .reachableFrom n,x .edgesBetween (x .reachableFrom n) is Relation-like NAT -defined Function-like finite [Graph-like] finite connected Component-like () inducedSubgraph of x,x .reachableFrom n,x .edgesBetween (x .reachableFrom n)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite connected Component-like () inducedSubgraph of x,x .reachableFrom n,x .edgesBetween (x .reachableFrom n) is non empty finite Element of bool (the_Vertices_of x)
the Relation-like NAT -defined Function-like finite [Graph-like] finite connected Component-like () inducedSubgraph of x,x .reachableFrom n,x .edgesBetween (x .reachableFrom n) . VertexSelector is set
C is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() set
max C is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative set
b is Element of the_Vertices_of G
{b} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
(G,{b}) is finite Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {b} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {b} & (G,b₁,b₂) ) ) } is set
(G,{b}) /\ H is finite Element of bool (the_Vertices_of G)
card ((G,{b}) /\ H) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
(x .reachableFrom n) /\ H is finite Element of bool (the_Vertices_of G)
G .edgesBetween H is finite Element of bool (the_Edges_of G)
G .edgesInto H is finite Element of bool (the_Edges_of G)
G .edgesOutOf H is finite Element of bool (the_Edges_of G)
(G .edgesInto H) /\ (G .edgesOutOf H) is finite Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,H,G .edgesBetween H is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,H,G .edgesBetween H
2 * 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() even V121() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() Element of NAT
(2 * 0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(the_Vertices_of G) \ H is finite Element of bool (the_Vertices_of G)
the_Vertices_of x is non empty finite Element of bool (the_Vertices_of G)
aa is set
a is non empty finite Element of bool (the_Vertices_of G)
G .edgesBetween a is finite Element of bool (the_Edges_of G)
G .edgesInto a is finite Element of bool (the_Edges_of G)
G .edgesOutOf a is finite Element of bool (the_Edges_of G)
(G .edgesInto a) /\ (G .edgesOutOf a) is finite Element of bool (the_Edges_of G)
bb is Element of the_Vertices_of G
{bb} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
a \/ {bb} is non empty finite Element of bool (the_Vertices_of G)
G .edgesBetween (a \/ {bb}) is finite Element of bool (the_Edges_of G)
G .edgesInto (a \/ {bb}) is finite Element of bool (the_Edges_of G)
G .edgesOutOf (a \/ {bb}) is finite Element of bool (the_Edges_of G)
(G .edgesInto (a \/ {bb})) /\ (G .edgesOutOf (a \/ {bb})) is finite Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}) is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}) is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}) . VertexSelector is set
aa is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a,G .edgesBetween a
the_Vertices_of aa is non empty finite Element of bool (the_Vertices_of G)
aa . VertexSelector is set
(G,(the_Vertices_of aa)) is finite Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in the_Vertices_of aa & ex b₂ being Element of the_Vertices_of G st
( b₂ in the_Vertices_of aa & (G,b₁,b₂) ) ) } is set
b3 is Element of the_Vertices_of G
the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}) is finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}) . EdgeSelector is set
(the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) is non empty finite set
x is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
e is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
b3 is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
b3 .first() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
b3 . 1 is set
b3 .last() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
len b3 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
b3 . (len b3) is set
P is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
P .first() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
P . 1 is set
P .last() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
len P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (len P) is set
2 * 2 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(2 * 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
5 - 2 is complex V25() integer ext-real set
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (1 + 2) is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (1 + 1) is set
P . 2 is set
- 2 is complex V25() integer ext-real non positive set
5 + (- 2) is complex V25() integer ext-real set
(len P) + (- 2) is complex V25() integer ext-real set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len P) - (2 * 1) is non empty complex V25() integer non even ext-real set
j is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . j is set
(len P) - 2 is complex V25() integer ext-real set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}) is non empty finite Element of bool (the_Vertices_of G)
d is Element of the_Vertices_of G
{d} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
(G,{d}) is finite Element of bool (the_Vertices_of G)
{ b₁ where b₁ is Element of the_Vertices_of G : ( not b₁ in {d} & ex b₂ being Element of the_Vertices_of G st
( b₂ in {d} & (G,b₁,b₂) ) ) } is set
(G,{d}) /\ H is finite Element of bool (the_Vertices_of G)
((len P) - 2) + 2 is complex V25() integer ext-real set
P . (((len P) - 2) + 2) is set
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . (j + 1) is set
card ((G,{d}) /\ H) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
x is set
dom P is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
x is Element of the_Vertices_of G
k is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
P . k is set
k is set
k is set
k is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
P . k is set
Q1 is set
k is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P .cut (k,j) is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
(the_Vertices_of G) \/ (the_Edges_of G) is non empty finite set
cc is set
Q is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len Q is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len Q) + k is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
cc is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
dom Q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
- 1 is complex V25() integer ext-real non positive set
1 + (- 1) is complex V25() integer ext-real set
cc + (- 1) is complex V25() integer ext-real set
cc - 1 is complex V25() integer even ext-real set
dd is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
k + dd is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
- 0 is empty Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V25() integer finite finite-yielding V32() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V100() ext-real non positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V141() V148() V149() V150() V151() set
cc + (- 0) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
xs is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len P) + (- 0) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len P) + (- 1) is complex V25() integer ext-real set
Q . cc is set
(len P) - 1 is complex V25() integer even ext-real set
((len P) - 1) - k is non empty complex V25() integer non even ext-real set
(((len P) - 1) - k) + k is complex V25() integer even ext-real set
dd + k is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
dd + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
Q . (dd + 1) is set
P . xs is set
k + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
ys is set
Q .first() is Element of the_Vertices_of G
Q . 1 is set
Q .last() is Element of the_Vertices_of G
Q . (len Q) is set
(P .cut (k,j)) .first() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
(P .cut (k,j)) . 1 is set
P . k is set
xs is set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,H,G .edgesBetween H is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,H,G .edgesBetween H . VertexSelector is set
xs is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,H,G .edgesBetween H
ys is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,H,G .edgesBetween H
ej is set
(P .cut (k,j)) .last() is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
len (P .cut (k,j)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(P .cut (k,j)) . (len (P .cut (k,j))) is set
R is set
Q .vertices() is finite Element of bool (the_Vertices_of G)
Q .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(Q .vertexSeq())) is finite Element of bool (the_Vertices_of G)
(P .cut (k,j)) .vertices() is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}))
bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) is non empty finite V32() set
(P .cut (k,j)) .vertexSeq() is Relation-like NAT -defined the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}) -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
K426((the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})),((P .cut (k,j)) .vertexSeq())) is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}))
P .cut (1,k) is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
len (P .cut (1,k)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (P .cut (1,k))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R is set
R is set
R is set
R is set
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
Q . C is set
C + (- 1) is complex V25() integer ext-real set
C - 1 is complex V25() integer even ext-real set
m is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
k + m is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C + (- 0) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
Q . (m + 1) is set
P . n is set
k + (C + (- 1)) is complex V25() integer ext-real set
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
k + C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P .vertices() is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}))
P .vertexSeq() is Relation-like NAT -defined the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}) -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})
K426((the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb})),(P .vertexSeq())) is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,a \/ {bb},G .edgesBetween (a \/ {bb}))
{(Q .last()),bb,x,(Q .first())} is non empty finite set
R is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of G
len R is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
R .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (R .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
R .vertices() is finite Element of bool (the_Vertices_of G)
R .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(R .vertexSeq())) is finite Element of bool (the_Vertices_of G)
R . 1 is set
R . 3 is set
R . 5 is set
R . 7 is set
Q .edges() is finite Element of bool (the_Edges_of G)
Q .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
K426((the_Edges_of G),(Q .edgeSeq())) is finite Element of bool (the_Edges_of G)
R .edges() is finite Element of bool (the_Edges_of G)
K426((the_Edges_of G),(R .edgeSeq())) is finite Element of bool (the_Edges_of G)
(Q .edges()) /\ (R .edges()) is finite Element of bool (the_Edges_of G)
C is set
m is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
R . m is set
R . (1 + 2) is set
R . (1 + 1) is set
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
3 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (3 + 2) is set
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (3 + 1) is set
5 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (5 + 2) is set
5 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (5 + 1) is set
C is set
{(Q .first()),(Q .last())} is non empty finite Element of bool (the_Vertices_of G)
(Q .vertices()) /\ (R .vertices()) is finite Element of bool (the_Vertices_of G)
(Q .vertices()) /\ (R .vertices()) is finite Element of bool (the_Vertices_of G)
(Q .vertices()) /\ (R .vertices()) is finite Element of bool (the_Vertices_of G)
(Q .vertices()) /\ (R .vertices()) is finite Element of bool (the_Vertices_of G)
C is set
C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
C + (- 1) is complex V25() integer ext-real set
C - 1 is complex V25() integer even ext-real set
m is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
k + m is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C + (- 0) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P . n is set
(len P) - k is complex V25() integer even ext-real set
((len P) - k) + (- 1) is complex V25() integer ext-real set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
Q . C is set
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
Q . (m + 1) is set
e is set
C is set
Q .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
len (Q .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
Q .append R is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
R .first() is Element of the_Vertices_of G
(len Q) + 6 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(Q .append R) . ((len Q) + 6) is set
6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (6 + 1) is set
(len Q) + 4 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(Q .append R) . ((len Q) + 4) is set
4 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R . (4 + 1) is set
(len Q) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(Q .append R) . ((len Q) + 2) is set
R . (2 + 1) is set
(Q .append R) .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(Q .append R) .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len ((Q .append R) .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(Q .length()) + 3 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
1 + 3 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
R .last() is Element of the_Vertices_of G
R . (len R) is set
len (Q .append R) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len (Q .append R)) - 2 is complex V25() integer ext-real set
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
m is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
m + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(Q .append R) . m is set
(Q .append R) . n is set
e is set
e is set
(len (Q .append R)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len (Q .append R)) + 1) + (- 1) is complex V25() integer ext-real set
(len Q) + (len R) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len Q) + (len R)) + (- 1) is complex V25() integer ext-real set
((len Q) + 2) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len Q) + (2 * 2) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len Q) + 4) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len Q) + (2 * 2) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m + 0 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
Q . m is set
Q . n is set
Q . m is set
(m + 2) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m + 4 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
Q . m is set
(m + 2) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
m + 4 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len Q) + 2) + 4 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(2 * 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
3 - 2 is complex V25() integer ext-real set
((2 * 0) + 1) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
Q . m is set
m1 is set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite () set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
x is finite (G,S,P)
(the_Vertices_of G) \ x is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty finite V32() set
G .edgesBetween ((the_Vertices_of G) \ x) is finite Element of bool (the_Edges_of G)
the_Edges_of G is finite set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty finite V32() set
G .edgesInto ((the_Vertices_of G) \ x) is finite Element of bool (the_Edges_of G)
G .edgesOutOf ((the_Vertices_of G) \ x) is finite Element of bool (the_Edges_of G)
(G .edgesInto ((the_Vertices_of G) \ x)) /\ (G .edgesOutOf ((the_Vertices_of G) \ x)) is finite Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,(the_Vertices_of G) \ x,G .edgesBetween ((the_Vertices_of G) \ x)
the_Vertices_of x is non empty finite set
x . VertexSelector is set
n is Element of the_Vertices_of x
x .reachableFrom n is non empty finite Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty finite V32() set
H is Element of the_Vertices_of G
y is Element of the_Vertices_of G
C is Element of the_Vertices_of G
G is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
S is Relation-like NAT -defined Function-like finite [Graph-like] finite set
S .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of S is non empty finite set
S . VertexSelector is set
len (the_Vertices_of S) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
P is Relation-like NAT -defined Function-like finite [Graph-like] finite non trivial () set
the_Vertices_of P is non empty finite set
P . VertexSelector is set
x is Element of the_Vertices_of P
x is Element of the_Vertices_of P
n is finite (P,x,x)
(the_Vertices_of P) \ n is finite Element of bool (the_Vertices_of P)
bool (the_Vertices_of P) is non empty finite V32() set
P .edgesBetween ((the_Vertices_of P) \ n) is finite Element of bool (the_Edges_of P)
the_Edges_of P is finite set
P . EdgeSelector is set
bool (the_Edges_of P) is non empty finite V32() set
P .edgesInto ((the_Vertices_of P) \ n) is finite Element of bool (the_Edges_of P)
P .edgesOutOf ((the_Vertices_of P) \ n) is finite Element of bool (the_Edges_of P)
(P .edgesInto ((the_Vertices_of P) \ n)) /\ (P .edgesOutOf ((the_Vertices_of P) \ n)) is finite Element of bool (the_Edges_of P)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) . VertexSelector is set
y is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y is non empty finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n))
bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n)) is non empty finite V32() set
C is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C is non empty finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n))
( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n is non empty finite set
P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n) is finite Element of bool (the_Edges_of P)
P .edgesInto (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n) is finite Element of bool (the_Edges_of P)
P .edgesOutOf (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n) is finite Element of bool (the_Edges_of P)
(P .edgesInto (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n)) /\ (P .edgesOutOf (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n)) is finite Element of bool (the_Edges_of P)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n) is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) is non empty finite Element of bool (the_Vertices_of P)
( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) /\ ( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n))
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n) .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n) is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n) . VertexSelector is set
len (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n) is non empty finite Element of bool (the_Vertices_of P)
( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n is non empty finite set
P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n) is finite Element of bool (the_Edges_of P)
P .edgesInto (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n) is finite Element of bool (the_Edges_of P)
P .edgesOutOf (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n) is finite Element of bool (the_Edges_of P)
(P .edgesInto (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n)) /\ (P .edgesOutOf (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n)) is finite Element of bool (the_Edges_of P)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n) is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n) .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n) is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n) . VertexSelector is set
len (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n)) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n) is non empty finite Element of bool (the_Vertices_of P)
P .edgesBetween n is finite Element of bool (the_Edges_of P)
P .edgesInto n is finite Element of bool (the_Edges_of P)
P .edgesOutOf n is finite Element of bool (the_Edges_of P)
(P .edgesInto n) /\ (P .edgesOutOf n) is finite Element of bool (the_Edges_of P)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,n,P .edgesBetween n is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,n,P .edgesBetween n
bg is set
a is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n)
aa is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n)
bb is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,n,P .edgesBetween n is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,n,P .edgesBetween n . VertexSelector is set
e is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,n,P .edgesBetween n
x is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,n,P .edgesBetween n
x is Element of the_Vertices_of P
e is Element of the_Vertices_of P
aa is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n)
aa is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom C) \/ n)
e is set
bb is Element of the_Vertices_of S
{bb} is non empty trivial finite 1 -element Element of bool (the_Vertices_of S)
bool (the_Vertices_of S) is non empty finite V32() set
(S,{bb}) is finite Element of bool (the_Vertices_of S)
{ b₁ where b₁ is Element of the_Vertices_of S : ( not b₁ in {bb} & ex b₂ being Element of the_Vertices_of S st
( b₂ in {bb} & (S,b₁,b₂) ) ) } is set
x is Element of the_Vertices_of S
x is set
bg is Element of the_Vertices_of S
bg is Element of the_Vertices_of S
a is set
aa is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n)
bb is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n)
e is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,n,P .edgesBetween n is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,n,P .edgesBetween n . VertexSelector is set
x is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,n,P .edgesBetween n
x is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,n,P .edgesBetween n
e is Element of the_Vertices_of P
b3 is Element of the_Vertices_of P
bb is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n)
bb is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n,P .edgesBetween (( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n) .reachableFrom y) \/ n)
x is set
e is Element of the_Vertices_of S
{e} is non empty trivial finite 1 -element Element of bool (the_Vertices_of S)
bool (the_Vertices_of S) is non empty finite V32() set
(S,{e}) is finite Element of bool (the_Vertices_of S)
{ b₁ where b₁ is Element of the_Vertices_of S : ( not b₁ in {e} & ex b₂ being Element of the_Vertices_of S st
( b₂ in {e} & (S,b₁,b₂) ) ) } is set
x is Element of the_Vertices_of S
e is set
a is Element of the_Vertices_of S
a is Element of the_Vertices_of S
e is set
aa is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n)
bb is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of P,(the_Vertices_of P) \ n,P .edgesBetween ((the_Vertices_of P) \ n)
G is Relation-like NAT -defined Function-like finite [Graph-like] finite non trivial () set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite () set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
the Element of the_Vertices_of G is Element of the_Vertices_of G
S is Element of the_Vertices_of G
P is Element of the_Vertices_of G
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng S is finite set
P is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the_Vertices_of G
rng P is finite set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
card (the_Vertices_of G) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of omega
S is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
rng S is non empty finite set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of S is non empty finite set
S . VertexSelector is set
P is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
rng P is non empty finite set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
P is Element of the_Vertices_of G
P .. S is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
rng S is non empty finite set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
P is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
(S,P) is finite set
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(P,(len S)) -cut S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((P,(len S)) -cut S) is finite set
bool (the_Vertices_of G) is non empty finite V32() set
(P,(len S)) -cut S is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the_Vertices_of G
rng ((P,(len S)) -cut S) is finite set
rng S is non empty finite set
x is set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
(G,S,P) is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty finite V32() set
(P,(len S)) -cut S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((P,(len S)) -cut S) is finite set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(P,(len S)) -cut S is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the_Vertices_of G
len ((P,(len S)) -cut S) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len ((P,(len S)) -cut S)) + P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(len S) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) + 1) - P is complex V25() integer ext-real set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () () set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is Element of the_Vertices_of G
<*S*> is non empty trivial Relation-like NAT -defined the_Vertices_of G -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the_Vertices_of G
P is Element of the_Vertices_of G
{P} is non empty trivial finite 1 -element Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty finite V32() set
rng <*S*> is non empty trivial finite 1 -element set
x is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
n is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(G,x,n) is finite Element of bool (the_Vertices_of G)
(n,(len x)) -cut x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((n,(len x)) -cut x) is finite set
G .edgesBetween (G,x,n) is finite Element of bool (the_Edges_of G)
the_Edges_of G is finite set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty finite V32() set
G .edgesInto (G,x,n) is finite Element of bool (the_Edges_of G)
G .edgesOutOf (G,x,n) is finite Element of bool (the_Edges_of G)
(G .edgesInto (G,x,n)) /\ (G .edgesOutOf (G,x,n)) is finite Element of bool (the_Edges_of G)
x . n is set
H is Relation-like NAT -defined Function-like finite [Graph-like] finite trivial connected () () inducedSubgraph of G,(G,x,n),G .edgesBetween (G,x,n)
the_Vertices_of H is non empty finite set
H . VertexSelector is set
y is Element of the_Vertices_of H
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
dom S is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
P is Element of the_Vertices_of G
n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
H is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
y is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
S . n is set
S . H is set
S . y is set
(G,S,n) is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty finite V32() set
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(n,(len S)) -cut S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((n,(len S)) -cut S) is finite set
G .edgesBetween (G,S,n) is finite Element of bool (the_Edges_of G)
the_Edges_of G is finite set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty finite V32() set
G .edgesInto (G,S,n) is finite Element of bool (the_Edges_of G)
G .edgesOutOf (G,S,n) is finite Element of bool (the_Edges_of G)
(G .edgesInto (G,S,n)) /\ (G .edgesOutOf (G,S,n)) is finite Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n) is Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n)
(n,(len S)) -cut S is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the_Vertices_of G
n - n is complex V25() integer ext-real set
(len S) - n is complex V25() integer ext-real set
len ((n,(len S)) -cut S) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
(len ((n,(len S)) -cut S)) + n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
((len ((n,(len S)) -cut S)) + n) - n is complex V25() integer ext-real set
(len S) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len S) + 1) - n is complex V25() integer ext-real set
((len S) - n) + 1 is complex V25() integer ext-real set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
dom ((n,(len S)) -cut S) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of bool NAT
((n,(len S)) -cut S) . (0 + 1) is set
n + 0 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
S . (n + 0) is set
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n) is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n) . VertexSelector is set
b is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
n + b is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(((len S) + 1) - n) + n is complex V25() integer ext-real set
((n,(len S)) -cut S) . (b + 1) is set
S . (n + b) is set
a is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n)
v is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n)
n1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
n + n1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
n1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((n,(len S)) -cut S) . (n1 + 1) is set
S . (n + n1) is set
bg is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n)
{a} is non empty trivial finite 1 -element Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n))
bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n)) is non empty finite V32() set
( the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n),{a}) is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n))
{ b₁ where b₁ is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n) : ( not b₁ in {a} & ex b₂ being Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n) st
( b₂ in {a} & ( the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,n),G .edgesBetween (G,S,n),b₁,b₂) ) ) } is set
a is set
rng S is non empty finite set
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
(G,S,P) is finite Element of bool (the_Vertices_of G)
(P,(len S)) -cut S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((P,(len S)) -cut S) is finite set
G .edgesBetween (G,S,P) is finite Element of bool (the_Edges_of G)
G .edgesInto (G,S,P) is finite Element of bool (the_Edges_of G)
G .edgesOutOf (G,S,P) is finite Element of bool (the_Edges_of G)
(G .edgesInto (G,S,P)) /\ (G .edgesOutOf (G,S,P)) is finite Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,P),G .edgesBetween (G,S,P)
the_Vertices_of x is non empty finite set
x . VertexSelector is set
x is Element of the_Vertices_of x
S . P is set
P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
(G,S,P) is finite Element of bool (the_Vertices_of G)
(P,(len S)) -cut S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((P,(len S)) -cut S) is finite set
G .edgesBetween (G,S,P) is finite Element of bool (the_Edges_of G)
G .edgesInto (G,S,P) is finite Element of bool (the_Edges_of G)
G .edgesOutOf (G,S,P) is finite Element of bool (the_Edges_of G)
(G .edgesInto (G,S,P)) /\ (G .edgesOutOf (G,S,P)) is finite Element of bool (the_Edges_of G)
x is Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,P),G .edgesBetween (G,S,P)
the_Vertices_of x is non empty finite set
x . VertexSelector is set
x is Element of the_Vertices_of x
S . P is set
the_Vertices_of x is non empty finite Element of bool (the_Vertices_of G)
H is Element of the_Vertices_of x
y is Element of the_Vertices_of x
n is Element of the_Vertices_of G
C is Element of the_Vertices_of G
a is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
S . a is set
y .. S is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
S . b is set
H .. S is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
C is Element of the_Vertices_of G
x is Element of the_Vertices_of G
x is Element of the_Vertices_of G
P is Element of the_Vertices_of G
n is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
H is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
y is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
S . n is set
S . H is set
S . y is set
C is Element of the_Vertices_of G
a is Element of the_Vertices_of G
C is Element of the_Vertices_of G
b is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
v is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
n1 is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real non negative set
S . b is set
S . v is set
S . n1 is set
G is Relation-like NAT -defined Function-like finite [Graph-like] finite () set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
P is Relation-like NAT -defined Function-like finite [Graph-like] finite set
P .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of P is non empty finite set
P . VertexSelector is set
len (the_Vertices_of P) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
the Element of the_Vertices_of P is Element of the_Vertices_of P
<* the Element of the_Vertices_of P*> is non empty trivial Relation-like NAT -defined the_Vertices_of P -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the_Vertices_of P
x is non empty Relation-like NAT -defined the_Vertices_of P -valued Function-like finite FinSequence-like FinSubsequence-like (P)
x is Relation-like NAT -defined Function-like finite [Graph-like] finite non trivial () set
the_Vertices_of x is non empty finite set
x . VertexSelector is set
x is Element of the_Vertices_of x
{x} is non empty trivial finite 1 -element set
(the_Vertices_of x) \ {x} is finite Element of bool (the_Vertices_of x)
bool (the_Vertices_of x) is non empty finite V32() set
x .edgesBetween ((the_Vertices_of x) \ {x}) is finite Element of bool (the_Edges_of x)
the_Edges_of x is finite set
x . EdgeSelector is set
bool (the_Edges_of x) is non empty finite V32() set
x .edgesInto ((the_Vertices_of x) \ {x}) is finite Element of bool (the_Edges_of x)
x .edgesOutOf ((the_Vertices_of x) \ {x}) is finite Element of bool (the_Edges_of x)
(x .edgesInto ((the_Vertices_of x) \ {x})) /\ (x .edgesOutOf ((the_Vertices_of x) \ {x})) is finite Element of bool (the_Edges_of x)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x})
<*x*> is non empty trivial Relation-like NAT -defined the_Vertices_of x -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the_Vertices_of x
{x} is non empty trivial finite 1 -element Element of bool (the_Vertices_of x)
(the_Vertices_of x) \ {x} is finite Element of bool (the_Vertices_of x)
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) is non empty finite Element of bool (the_Vertices_of x)
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) . VertexSelector is set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) is non empty finite set
len (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x})) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
card ((the_Vertices_of x) \ {x}) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of omega
x .order() is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
len (the_Vertices_of x) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
card {x} is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of omega
(x .order()) - (card {x}) is complex V25() integer ext-real set
S - 1 is complex V25() integer ext-real set
H is non empty Relation-like NAT -defined the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) -valued Function-like finite FinSequence-like FinSubsequence-like ( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}))
rng H is non empty finite set
{x} /\ (rng H) is finite Element of bool (the_Vertices_of x)
{x} /\ ((the_Vertices_of x) \ {x}) is finite Element of bool (the_Vertices_of x)
{x} /\ (the_Vertices_of x) is finite Element of bool (the_Vertices_of x)
({x} /\ (the_Vertices_of x)) \ {x} is finite Element of bool (the_Vertices_of x)
{x} \ {x} is finite Element of bool (the_Vertices_of x)
rng <*x*> is non empty trivial finite 1 -element set
<*x*> ^ H is non empty Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng (<*x*> ^ H) is non empty finite set
((the_Vertices_of x) \ {x}) \/ (rng <*x*>) is non empty finite set
((the_Vertices_of x) \ {x}) \/ {x} is non empty finite Element of bool (the_Vertices_of x)
{x} \/ (the_Vertices_of x) is non empty finite set
C is Relation-like NAT -defined the_Vertices_of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the_Vertices_of x
C is non empty Relation-like NAT -defined the_Vertices_of P -valued Function-like finite FinSequence-like FinSubsequence-like (P)
a is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
len C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(P,C,a) is finite Element of bool (the_Vertices_of P)
bool (the_Vertices_of P) is non empty finite V32() set
(a,(len C)) -cut C is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((a,(len C)) -cut C) is finite set
P .edgesBetween (P,C,a) is finite Element of bool (the_Edges_of P)
the_Edges_of P is finite set
P . EdgeSelector is set
bool (the_Edges_of P) is non empty finite V32() set
P .edgesInto (P,C,a) is finite Element of bool (the_Edges_of P)
P .edgesOutOf (P,C,a) is finite Element of bool (the_Edges_of P)
(P .edgesInto (P,C,a)) /\ (P .edgesOutOf (P,C,a)) is finite Element of bool (the_Edges_of P)
C . a is set
b is Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of P,(P,C,a),P .edgesBetween (P,C,a)
the_Vertices_of b is non empty finite set
b . VertexSelector is set
v is Element of the_Vertices_of b
rng C is non empty finite set
- 1 is complex V25() integer ext-real non positive set
a + (- 1) is complex V25() integer ext-real set
1 + (- 1) is complex V25() integer ext-real set
a - 1 is complex V25() integer ext-real set
len <*x*> is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
n1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
H . n1 is set
( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}),H,n1) is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}))
bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x})) is non empty finite V32() set
len H is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(n1,(len H)) -cut H is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((n1,(len H)) -cut H) is finite set
n1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(P,C,(n1 + 1)) is finite Element of bool (the_Vertices_of P)
((n1 + 1),(len C)) -cut C is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng (((n1 + 1),(len C)) -cut C) is finite set
(len C) + (- 1) is complex V25() integer ext-real set
(len <*x*>) + (len H) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((len <*x*>) + (len H)) + (- 1) is complex V25() integer ext-real set
1 + (len H) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(1 + (len H)) + (- 1) is complex V25() integer ext-real set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) .edgesBetween ( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}),H,n1) is finite Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}))
the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) is finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) . EdgeSelector is set
bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x})) is non empty finite V32() set
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) .edgesInto ( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}),H,n1) is finite Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}))
the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) .edgesOutOf ( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}),H,n1) is finite Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}))
( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) .edgesInto ( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}),H,n1)) /\ ( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}) .edgesOutOf ( the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}),H,n1)) is finite Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of x,(the_Vertices_of x) \ {x},x .edgesBetween ((the_Vertices_of x) \ {x}))
S is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
G is Relation-like NAT -defined Function-like finite [Graph-like] finite () set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is Relation-like NAT -defined Function-like finite [Graph-like] finite () set
the_Vertices_of S is non empty finite set
S . VertexSelector is set
P is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G) (G)
x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal ext-real positive non negative set
x is non empty Relation-like NAT -defined the_Vertices_of S -valued Function-like finite FinSequence-like FinSubsequence-like (S)
len x is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(S,x,x) is finite Element of bool (the_Vertices_of S)
bool (the_Vertices_of S) is non empty finite V32() set
(x,(len x)) -cut x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((x,(len x)) -cut x) is finite set
S .edgesBetween (S,x,x) is finite Element of bool (the_Edges_of S)
the_Edges_of S is finite set
S . EdgeSelector is set
bool (the_Edges_of S) is non empty finite V32() set
S .edgesInto (S,x,x) is finite Element of bool (the_Edges_of S)
S .edgesOutOf (S,x,x) is finite Element of bool (the_Edges_of S)
(S .edgesInto (S,x,x)) /\ (S .edgesOutOf (S,x,x)) is finite Element of bool (the_Edges_of S)
n is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of S,(S,x,x),S .edgesBetween (S,x,x)
the_Vertices_of n is non empty finite set
n . VertexSelector is set
H is Element of the_Vertices_of n
x . x is set
(G,P,x) is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty finite V32() set
len P is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(x,(len P)) -cut P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng ((x,(len P)) -cut P) is finite set
G .edgesBetween (G,P,x) is finite Element of bool (the_Edges_of G)
the_Edges_of G is finite set
G . EdgeSelector is set
bool (the_Edges_of G) is non empty finite V32() set
G .edgesInto (G,P,x) is finite Element of bool (the_Edges_of G)
G .edgesOutOf (G,P,x) is finite Element of bool (the_Edges_of G)
(G .edgesInto (G,P,x)) /\ (G .edgesOutOf (G,P,x)) is finite Element of bool (the_Edges_of G)
S .edgesBetween (G,P,x) is finite Element of bool (the_Edges_of S)
S .edgesInto (G,P,x) is finite Element of bool (the_Edges_of S)
S .edgesOutOf (G,P,x) is finite Element of bool (the_Edges_of S)
(S .edgesInto (G,P,x)) /\ (S .edgesOutOf (G,P,x)) is finite Element of bool (the_Edges_of S)
y is Relation-like NAT -defined Function-like finite [Graph-like] finite () inducedSubgraph of G,(G,P,x),G .edgesBetween (G,P,x)
the_Vertices_of y is non empty finite set
y . VertexSelector is set
C is Element of the_Vertices_of y
G is Relation-like NAT -defined Function-like finite [Graph-like] finite set
the_Vertices_of G is non empty finite set
G . VertexSelector is set
S is non empty Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like (G)
rng S is non empty finite set
the_Edges_of G is finite set
G . EdgeSelector is set
(the_Vertices_of G) \/ (the_Edges_of G) is non empty finite set
P is Relation-like NAT -defined (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of G
P .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
P .edgeSeq() is Relation-like NAT -defined the_Edges_of G -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of G
len (P .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
P .vertices() is finite Element of bool (the_Vertices_of G)
bool (the_Vertices_of G) is non empty finite V32() set
P .vertexSeq() is Relation-like NAT -defined the_Vertices_of G -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of G
K426((the_Vertices_of G),(P .vertexSeq())) is finite Element of bool (the_Vertices_of G)
x is set
x .. S is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
x is Element of the_Vertices_of G
(G,S,x) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
(G,S,(G,S,x)) is finite Element of bool (the_Vertices_of G)
len S is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
((G,S,x),(len S)) -cut S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng (((G,S,x),(len S)) -cut S) is finite set
G .edgesBetween (G,S,(G,S,x)) is finite Element of bool (the_Edges_of G)
bool (the_Edges_of G) is non empty finite V32() set
G .edgesInto (G,S,(G,S,x)) is finite Element of bool (the_Edges_of G)
G .edgesOutOf (G,S,(G,S,x)) is finite Element of bool (the_Edges_of G)
(G .edgesInto (G,S,(G,S,x))) /\ (G .edgesOutOf (G,S,(G,S,x))) is finite Element of bool (the_Edges_of G)
the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)) is Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))
dom S is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of bool NAT
the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)) is non empty finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)) . VertexSelector is set
C is set
C is Element of the_Vertices_of G
(G,S,C) is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)) is finite set
the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)) . EdgeSelector is set
(the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))) is non empty finite set
C is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))) -valued Function-like finite FinSequence-like FinSubsequence-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))
C is Relation-like NAT -defined (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))) \/ (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Walk of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))
y is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))
C .vertices() is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)))
bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))) is non empty finite V32() set
C .vertexSeq() is Relation-like NAT -defined the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)) -valued Function-like finite FinSequence-like FinSubsequence-like VertexSeq of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))
K426((the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))),(C .vertexSeq())) is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)))
S . (G,S,x) is set
C .length() is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
C .edgeSeq() is Relation-like NAT -defined the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)) -valued Function-like finite FinSequence-like FinSubsequence-like EdgeSeq of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))
len (C .edgeSeq()) is epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal V121() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V148() V149() V150() Element of NAT
len C is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
{y} is non empty trivial finite 1 -element Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)))
( the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)),{y}) is finite Element of bool (the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)))
{ b₁ where b₁ is Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)) : ( not b₁ in {y} & ex b₂ being Element of the_Vertices_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)) st
( b₂ in {y} & ( the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)),b₁,b₂) ) ) } is set
C .edges() is finite Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)))
bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))) is non empty finite V32() set
K426((the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x))),(C .edgeSeq())) is finite Element of bool (the_Edges_of the Relation-like NAT -defined Function-like finite [Graph-like] finite inducedSubgraph of G,(G,S,(G,S,x)),G .edgesBetween (G,S,(G,S,x)))
b is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
a is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even ext-real positive non negative set
a + 2 is non empty epsilon-transitive epsilon-connected ordinal natural complex V25() integer finite cardinal non even V121() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() V149() V150() Element of NAT
C . a is set
C . b is set
v is set