:: TMAP_1 semantic presentation
begin
registration
let X be non empty TopSpace;
let X1, X2 be non empty SubSpace of X;
clusterX1 union X2 -> TopSpace-like ;
coherence
X1 union X2 is TopSpace-like ;
end;
definition
let A, B be non empty set ;
let A1, A2 be non empty Subset of A;
let f1 be Function of A1,B;
let f2 be Function of A2,B;
assume A1: f1 | (A1 /\ A2) = f2 | (A1 /\ A2) ;
funcf1 union f2 -> Function of (A1 \/ A2),B means :Def1: :: TMAP_1:def 1
( it | A1 = f1 & it | A2 = f2 );
existence
ex b1 being Function of (A1 \/ A2),B st
( b1 | A1 = f1 & b1 | A2 = f2 )
proof
reconsider A0 = A1 \/ A2 as non empty Subset of A ;
set G = f2 +* f1;
set H = f1 +* f2;
rng (f1 +* f2) c= (rng f1) \/ (rng f2) by FUNCT_4:17;
then A2: rng (f1 +* f2) c= B by XBOOLE_1:1;
rng (f2 +* f1) c= (rng f2) \/ (rng f1) by FUNCT_4:17;
then A3: rng (f2 +* f1) c= B by XBOOLE_1:1;
A4: ( dom f1 = A1 & dom f2 = A2 ) by FUNCT_2:def_1;
then dom (f1 +* f2) = A1 \/ A2 by FUNCT_4:def_1;
then reconsider F0 = f1 +* f2 as Function of (A1 \/ A2),B by A2, FUNCT_2:def_1, RELSET_1:4;
dom (f2 +* f1) = A1 \/ A2 by A4, FUNCT_4:def_1;
then reconsider F1 = f2 +* f1 as Function of (A1 \/ A2),B by A3, FUNCT_2:def_1, RELSET_1:4;
take F0 ; ::_thesis: ( F0 | A1 = f1 & F0 | A2 = f2 )
A5: now__::_thesis:_for_a_being_Element_of_A0_holds_
(_(_a_in_A2_implies_(f1_+*_f2)_._a_=_f2_._a_)_&_(_a_in_A1_\_A2_implies_(f1_+*_f2)_._a_=_f1_._a_)_)
let a be Element of A0; ::_thesis: ( ( a in A2 implies (f1 +* f2) . a = f2 . a ) & ( a in A1 \ A2 implies (f1 +* f2) . a = f1 . a ) )
thus ( a in A2 implies (f1 +* f2) . a = f2 . a ) by A4, FUNCT_4:def_1; ::_thesis: ( a in A1 \ A2 implies (f1 +* f2) . a = f1 . a )
thus ( a in A1 \ A2 implies (f1 +* f2) . a = f1 . a ) ::_thesis: verum
proof
assume a in A1 \ A2 ; ::_thesis: (f1 +* f2) . a = f1 . a
then not a in A2 by XBOOLE_0:def_5;
hence (f1 +* f2) . a = f1 . a by A4, FUNCT_4:def_1; ::_thesis: verum
end;
end;
A6: now__::_thesis:_for_a_being_Element_of_A0_holds_
(_(_a_in_A1_implies_(f2_+*_f1)_._a_=_f1_._a_)_&_(_a_in_A2_\_A1_implies_(f2_+*_f1)_._a_=_f2_._a_)_)
let a be Element of A0; ::_thesis: ( ( a in A1 implies (f2 +* f1) . a = f1 . a ) & ( a in A2 \ A1 implies (f2 +* f1) . a = f2 . a ) )
thus ( a in A1 implies (f2 +* f1) . a = f1 . a ) by A4, FUNCT_4:def_1; ::_thesis: ( a in A2 \ A1 implies (f2 +* f1) . a = f2 . a )
thus ( a in A2 \ A1 implies (f2 +* f1) . a = f2 . a ) ::_thesis: verum
proof
assume a in A2 \ A1 ; ::_thesis: (f2 +* f1) . a = f2 . a
then not a in A1 by XBOOLE_0:def_5;
hence (f2 +* f1) . a = f2 . a by A4, FUNCT_4:def_1; ::_thesis: verum
end;
end;
A7: now__::_thesis:_for_a_being_Element_of_A0_holds_F0_._a_=_F1_._a
let a be Element of A0; ::_thesis: F0 . a = F1 . a
A8: now__::_thesis:_(_a_in_A1_/\_A2_implies_F0_._a_=_F1_._a_)
assume A9: a in A1 /\ A2 ; ::_thesis: F0 . a = F1 . a
then A10: a in A1 by XBOOLE_0:def_4;
a in A2 by A9, XBOOLE_0:def_4;
then A11: F0 . a = f2 . a by A5;
f1 . a = (f2 | (A1 /\ A2)) . a by A1, A9, FUNCT_1:49
.= f2 . a by A9, FUNCT_1:49 ;
hence F0 . a = F1 . a by A6, A10, A11; ::_thesis: verum
end;
A12: now__::_thesis:_(_a_in_A1_\+\_A2_implies_F0_._a_=_F1_._a_)
A13: now__::_thesis:_(_a_in_A2_\_A1_implies_F0_._a_=_F1_._a_)
assume A14: a in A2 \ A1 ; ::_thesis: F0 . a = F1 . a
A2 \ A1 c= A2 by XBOOLE_1:36;
hence F0 . a = f2 . a by A5, A14
.= F1 . a by A6, A14 ;
::_thesis: verum
end;
A15: now__::_thesis:_(_a_in_A1_\_A2_implies_F0_._a_=_F1_._a_)
A16: A1 \ A2 c= A1 by XBOOLE_1:36;
assume A17: a in A1 \ A2 ; ::_thesis: F0 . a = F1 . a
hence F0 . a = f1 . a by A5
.= F1 . a by A6, A17, A16 ;
::_thesis: verum
end;
assume a in A1 \+\ A2 ; ::_thesis: F0 . a = F1 . a
hence F0 . a = F1 . a by A15, A13, XBOOLE_0:def_3; ::_thesis: verum
end;
A0 = (A1 \+\ A2) \/ (A1 /\ A2) by XBOOLE_1:93;
hence F0 . a = F1 . a by A12, A8, XBOOLE_0:def_3; ::_thesis: verum
end;
now__::_thesis:_(_A1_is_non_empty_Subset_of_A0_&_(_for_a_being_Element_of_A0_st_a_in_A1_holds_
F0_._a_=_f1_._a_)_)
thus A1 is non empty Subset of A0 by XBOOLE_1:7; ::_thesis: for a being Element of A0 st a in A1 holds
F0 . a = f1 . a
let a be Element of A0; ::_thesis: ( a in A1 implies F0 . a = f1 . a )
assume A18: a in A1 ; ::_thesis: F0 . a = f1 . a
thus F0 . a = F1 . a by A7
.= f1 . a by A6, A18 ; ::_thesis: verum
end;
hence F0 | A1 = f1 by FUNCT_2:96; ::_thesis: F0 | A2 = f2
A2 is non empty Subset of A0 by XBOOLE_1:7;
hence F0 | A2 = f2 by A5, FUNCT_2:96; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (A1 \/ A2),B st b1 | A1 = f1 & b1 | A2 = f2 & b2 | A1 = f1 & b2 | A2 = f2 holds
b1 = b2
proof
reconsider A0 = A1 \/ A2 as non empty Subset of A ;
let F, G be Function of (A1 \/ A2),B; ::_thesis: ( F | A1 = f1 & F | A2 = f2 & G | A1 = f1 & G | A2 = f2 implies F = G )
assume that
A19: F | A1 = f1 and
A20: F | A2 = f2 and
A21: G | A1 = f1 and
A22: G | A2 = f2 ; ::_thesis: F = G
now__::_thesis:_for_a_being_Element_of_A0_holds_F_._a_=_G_._a
let a be Element of A0; ::_thesis: F . a = G . a
A23: now__::_thesis:_(_a_in_A2_implies_F_._a_=_G_._a_)
assume A24: a in A2 ; ::_thesis: F . a = G . a
hence F . a = (G | A2) . a by A20, A22, FUNCT_1:49
.= G . a by A24, FUNCT_1:49 ;
::_thesis: verum
end;
now__::_thesis:_(_a_in_A1_implies_F_._a_=_G_._a_)
assume A25: a in A1 ; ::_thesis: F . a = G . a
hence F . a = (G | A1) . a by A19, A21, FUNCT_1:49
.= G . a by A25, FUNCT_1:49 ;
::_thesis: verum
end;
hence F . a = G . a by A23, XBOOLE_0:def_3; ::_thesis: verum
end;
hence F = G by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines union TMAP_1:def_1_:_
for A, B being non empty set
for A1, A2 being non empty Subset of A
for f1 being Function of A1,B
for f2 being Function of A2,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) holds
for b7 being Function of (A1 \/ A2),B holds
( b7 = f1 union f2 iff ( b7 | A1 = f1 & b7 | A2 = f2 ) );
theorem Th1: :: TMAP_1:1
for A, B being non empty set
for A1, A2 being non empty Subset of A st A1 misses A2 holds
for f1 being Function of A1,B
for f2 being Function of A2,B holds
( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 )
proof
let A, B be non empty set ; ::_thesis: for A1, A2 being non empty Subset of A st A1 misses A2 holds
for f1 being Function of A1,B
for f2 being Function of A2,B holds
( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 )
let A1, A2 be non empty Subset of A; ::_thesis: ( A1 misses A2 implies for f1 being Function of A1,B
for f2 being Function of A2,B holds
( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 ) )
assume A1 misses A2 ; ::_thesis: for f1 being Function of A1,B
for f2 being Function of A2,B holds
( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 )
then A1: A1 /\ A2 = {} by XBOOLE_0:def_7;
let f1 be Function of A1,B; ::_thesis: for f2 being Function of A2,B holds
( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 )
let f2 be Function of A2,B; ::_thesis: ( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 )
A1 /\ A2 c= A2 by XBOOLE_1:17;
then reconsider g2 = f2 | (A1 /\ A2) as Function of {},B by A1, FUNCT_2:32;
A1 /\ A2 c= A1 by XBOOLE_1:17;
then reconsider g1 = f1 | (A1 /\ A2) as Function of {},B by A1, FUNCT_2:32;
g1 = g2 ;
hence ( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 ) by Def1; ::_thesis: verum
end;
theorem Th2: :: TMAP_1:2
for A, B being non empty set
for A1, A2 being non empty Subset of A
for g being Function of (A1 \/ A2),B
for g1 being Function of A1,B
for g2 being Function of A2,B st g | A1 = g1 & g | A2 = g2 holds
g = g1 union g2
proof
let A, B be non empty set ; ::_thesis: for A1, A2 being non empty Subset of A
for g being Function of (A1 \/ A2),B
for g1 being Function of A1,B
for g2 being Function of A2,B st g | A1 = g1 & g | A2 = g2 holds
g = g1 union g2
let A1, A2 be non empty Subset of A; ::_thesis: for g being Function of (A1 \/ A2),B
for g1 being Function of A1,B
for g2 being Function of A2,B st g | A1 = g1 & g | A2 = g2 holds
g = g1 union g2
let g be Function of (A1 \/ A2),B; ::_thesis: for g1 being Function of A1,B
for g2 being Function of A2,B st g | A1 = g1 & g | A2 = g2 holds
g = g1 union g2
let g1 be Function of A1,B; ::_thesis: for g2 being Function of A2,B st g | A1 = g1 & g | A2 = g2 holds
g = g1 union g2
let g2 be Function of A2,B; ::_thesis: ( g | A1 = g1 & g | A2 = g2 implies g = g1 union g2 )
assume A1: ( g | A1 = g1 & g | A2 = g2 ) ; ::_thesis: g = g1 union g2
A2 c= A1 \/ A2 by XBOOLE_1:7;
then reconsider f2 = g | A2 as Function of A2,B by FUNCT_2:32;
A1 c= A1 \/ A2 by XBOOLE_1:7;
then reconsider f1 = g | A1 as Function of A1,B by FUNCT_2:32;
A2: A1 /\ A2 c= A2 by XBOOLE_1:17;
A1 /\ A2 c= A1 by XBOOLE_1:17;
then f1 | (A1 /\ A2) = g | (A1 /\ A2) by FUNCT_1:51
.= f2 | (A1 /\ A2) by A2, FUNCT_1:51 ;
hence g = g1 union g2 by A1, Def1; ::_thesis: verum
end;
theorem :: TMAP_1:3
for A, B being non empty set
for A1, A2 being non empty Subset of A
for f1 being Function of A1,B
for f2 being Function of A2,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) holds
f1 union f2 = f2 union f1
proof
let A, B be non empty set ; ::_thesis: for A1, A2 being non empty Subset of A
for f1 being Function of A1,B
for f2 being Function of A2,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) holds
f1 union f2 = f2 union f1
let A1, A2 be non empty Subset of A; ::_thesis: for f1 being Function of A1,B
for f2 being Function of A2,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) holds
f1 union f2 = f2 union f1
let f1 be Function of A1,B; ::_thesis: for f2 being Function of A2,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) holds
f1 union f2 = f2 union f1
let f2 be Function of A2,B; ::_thesis: ( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) implies f1 union f2 = f2 union f1 )
assume f1 | (A1 /\ A2) = f2 | (A1 /\ A2) ; ::_thesis: f1 union f2 = f2 union f1
then ( (f1 union f2) | A1 = f1 & (f1 union f2) | A2 = f2 ) by Def1;
hence f1 union f2 = f2 union f1 by Th2; ::_thesis: verum
end;
theorem :: TMAP_1:4
for A, B being non empty set
for A1, A2, A3, A12, A23 being non empty Subset of A st A12 = A1 \/ A2 & A23 = A2 \/ A3 holds
for f1 being Function of A1,B
for f2 being Function of A2,B
for f3 being Function of A3,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & f2 | (A2 /\ A3) = f3 | (A2 /\ A3) & f1 | (A1 /\ A3) = f3 | (A1 /\ A3) holds
for f12 being Function of A12,B
for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23
proof
let A, B be non empty set ; ::_thesis: for A1, A2, A3, A12, A23 being non empty Subset of A st A12 = A1 \/ A2 & A23 = A2 \/ A3 holds
for f1 being Function of A1,B
for f2 being Function of A2,B
for f3 being Function of A3,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & f2 | (A2 /\ A3) = f3 | (A2 /\ A3) & f1 | (A1 /\ A3) = f3 | (A1 /\ A3) holds
for f12 being Function of A12,B
for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23
let A1, A2, A3 be non empty Subset of A; ::_thesis: for A12, A23 being non empty Subset of A st A12 = A1 \/ A2 & A23 = A2 \/ A3 holds
for f1 being Function of A1,B
for f2 being Function of A2,B
for f3 being Function of A3,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & f2 | (A2 /\ A3) = f3 | (A2 /\ A3) & f1 | (A1 /\ A3) = f3 | (A1 /\ A3) holds
for f12 being Function of A12,B
for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23
let A12, A23 be non empty Subset of A; ::_thesis: ( A12 = A1 \/ A2 & A23 = A2 \/ A3 implies for f1 being Function of A1,B
for f2 being Function of A2,B
for f3 being Function of A3,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & f2 | (A2 /\ A3) = f3 | (A2 /\ A3) & f1 | (A1 /\ A3) = f3 | (A1 /\ A3) holds
for f12 being Function of A12,B
for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23 )
assume that
A1: A12 = A1 \/ A2 and
A2: A23 = A2 \/ A3 ; ::_thesis: for f1 being Function of A1,B
for f2 being Function of A2,B
for f3 being Function of A3,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & f2 | (A2 /\ A3) = f3 | (A2 /\ A3) & f1 | (A1 /\ A3) = f3 | (A1 /\ A3) holds
for f12 being Function of A12,B
for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23
let f1 be Function of A1,B; ::_thesis: for f2 being Function of A2,B
for f3 being Function of A3,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & f2 | (A2 /\ A3) = f3 | (A2 /\ A3) & f1 | (A1 /\ A3) = f3 | (A1 /\ A3) holds
for f12 being Function of A12,B
for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23
let f2 be Function of A2,B; ::_thesis: for f3 being Function of A3,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & f2 | (A2 /\ A3) = f3 | (A2 /\ A3) & f1 | (A1 /\ A3) = f3 | (A1 /\ A3) holds
for f12 being Function of A12,B
for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23
let f3 be Function of A3,B; ::_thesis: ( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) & f2 | (A2 /\ A3) = f3 | (A2 /\ A3) & f1 | (A1 /\ A3) = f3 | (A1 /\ A3) implies for f12 being Function of A12,B
for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23 )
assume that
A3: f1 | (A1 /\ A2) = f2 | (A1 /\ A2) and
A4: f2 | (A2 /\ A3) = f3 | (A2 /\ A3) and
A5: f1 | (A1 /\ A3) = f3 | (A1 /\ A3) ; ::_thesis: for f12 being Function of A12,B
for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23
let f12 be Function of A12,B; ::_thesis: for f23 being Function of A23,B st f12 = f1 union f2 & f23 = f2 union f3 holds
f12 union f3 = f1 union f23
let f23 be Function of A23,B; ::_thesis: ( f12 = f1 union f2 & f23 = f2 union f3 implies f12 union f3 = f1 union f23 )
assume that
A6: f12 = f1 union f2 and
A7: f23 = f2 union f3 ; ::_thesis: f12 union f3 = f1 union f23
A8: (f12 | A2) | (A2 /\ A3) = f2 | (A2 /\ A3) by A3, A6, Def1;
A1 \/ A23 = A12 \/ A3 by A1, A2, XBOOLE_1:4;
then reconsider f = f12 union f3 as Function of (A1 \/ A23),B ;
A12 /\ A3 c= A12 by XBOOLE_1:17;
then reconsider F = f12 | (A12 /\ A3) as Function of (A12 /\ A3),B by FUNCT_2:32;
A9: A2 /\ A3 c= A2 by XBOOLE_1:17;
A10: f12 | A2 = f2 by A3, A6, Def1;
A23 c= A1 \/ A23 by XBOOLE_1:7;
then reconsider H = f | A23 as Function of A23,B by FUNCT_2:32;
A11: A2 c= A12 by A1, XBOOLE_1:7;
A12 /\ A3 c= A3 by XBOOLE_1:17;
then reconsider G = f3 | (A12 /\ A3) as Function of (A12 /\ A3),B by FUNCT_2:32;
A12: A1 /\ A3 c= A1 by XBOOLE_1:17;
A13: (f12 | A1) | (A1 /\ A3) = f1 | (A1 /\ A3) by A3, A6, Def1;
now__::_thesis:_for_x_being_set_st_x_in_A12_/\_A3_holds_
F_._x_=_G_._x
let x be set ; ::_thesis: ( x in A12 /\ A3 implies F . x = G . x )
assume A14: x in A12 /\ A3 ; ::_thesis: F . x = G . x
A15: A1 /\ A3 c= A12 /\ A3 by A1, XBOOLE_1:7, XBOOLE_1:26;
A16: now__::_thesis:_(_x_in_A1_/\_A3_implies_F_._x_=_G_._x_)
assume A17: x in A1 /\ A3 ; ::_thesis: F . x = G . x
hence F . x = (F | (A1 /\ A3)) . x by FUNCT_1:49
.= (f12 | (A1 /\ A3)) . x by A15, FUNCT_1:51
.= (f3 | (A1 /\ A3)) . x by A5, A13, A12, FUNCT_1:51
.= (G | (A1 /\ A3)) . x by A15, FUNCT_1:51
.= G . x by A17, FUNCT_1:49 ;
::_thesis: verum
end;
A18: A2 /\ A3 c= A12 /\ A3 by A1, XBOOLE_1:7, XBOOLE_1:26;
A19: now__::_thesis:_(_x_in_A2_/\_A3_implies_F_._x_=_G_._x_)
assume A20: x in A2 /\ A3 ; ::_thesis: F . x = G . x
hence F . x = (F | (A2 /\ A3)) . x by FUNCT_1:49
.= (f12 | (A2 /\ A3)) . x by A18, FUNCT_1:51
.= (f3 | (A2 /\ A3)) . x by A4, A8, A9, FUNCT_1:51
.= (G | (A2 /\ A3)) . x by A18, FUNCT_1:51
.= G . x by A20, FUNCT_1:49 ;
::_thesis: verum
end;
A12 /\ A3 = (A1 /\ A3) \/ (A2 /\ A3) by A1, XBOOLE_1:23;
hence F . x = G . x by A14, A16, A19, XBOOLE_0:def_3; ::_thesis: verum
end;
then A21: f12 | (A12 /\ A3) = f3 | (A12 /\ A3) by FUNCT_2:12;
then A22: (f | A12) | A1 = f12 | A1 by Def1;
(f | A12) | A2 = f12 | A2 by A21, Def1;
then A23: f | A2 = f2 by A10, A11, FUNCT_1:51;
now__::_thesis:_for_x_being_set_st_x_in_A23_holds_
H_._x_=_f23_._x
let x be set ; ::_thesis: ( x in A23 implies H . x = f23 . x )
assume A24: x in A23 ; ::_thesis: H . x = f23 . x
A25: now__::_thesis:_(_x_in_A2_implies_H_._x_=_f23_._x_)
assume A26: x in A2 ; ::_thesis: H . x = f23 . x
thus H . x = f . x by A24, FUNCT_1:49
.= f2 . x by A23, A26, FUNCT_1:49
.= (f23 | A2) . x by A4, A7, Def1
.= f23 . x by A26, FUNCT_1:49 ; ::_thesis: verum
end;
now__::_thesis:_(_x_in_A3_implies_H_._x_=_f23_._x_)
assume A27: x in A3 ; ::_thesis: H . x = f23 . x
thus H . x = f . x by A24, FUNCT_1:49
.= (f | A3) . x by A27, FUNCT_1:49
.= f3 . x by A21, Def1
.= (f23 | A3) . x by A4, A7, Def1
.= f23 . x by A27, FUNCT_1:49 ; ::_thesis: verum
end;
hence H . x = f23 . x by A2, A24, A25, XBOOLE_0:def_3; ::_thesis: verum
end;
then A28: f | A23 = f23 by FUNCT_2:12;
A29: A1 c= A12 by A1, XBOOLE_1:7;
f12 | A1 = f1 by A3, A6, Def1;
then f | A1 = f1 by A22, A29, FUNCT_1:51;
hence f12 union f3 = f1 union f23 by A28, Th2; ::_thesis: verum
end;
theorem :: TMAP_1:5
for A, B being non empty set
for A1, A2 being non empty Subset of A
for f1 being Function of A1,B
for f2 being Function of A2,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) holds
( ( A1 is Subset of A2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies A1 is Subset of A2 ) & ( A2 is Subset of A1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies A2 is Subset of A1 ) )
proof
let A, B be non empty set ; ::_thesis: for A1, A2 being non empty Subset of A
for f1 being Function of A1,B
for f2 being Function of A2,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) holds
( ( A1 is Subset of A2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies A1 is Subset of A2 ) & ( A2 is Subset of A1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies A2 is Subset of A1 ) )
let A1, A2 be non empty Subset of A; ::_thesis: for f1 being Function of A1,B
for f2 being Function of A2,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) holds
( ( A1 is Subset of A2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies A1 is Subset of A2 ) & ( A2 is Subset of A1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies A2 is Subset of A1 ) )
let f1 be Function of A1,B; ::_thesis: for f2 being Function of A2,B st f1 | (A1 /\ A2) = f2 | (A1 /\ A2) holds
( ( A1 is Subset of A2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies A1 is Subset of A2 ) & ( A2 is Subset of A1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies A2 is Subset of A1 ) )
let f2 be Function of A2,B; ::_thesis: ( f1 | (A1 /\ A2) = f2 | (A1 /\ A2) implies ( ( A1 is Subset of A2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies A1 is Subset of A2 ) & ( A2 is Subset of A1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies A2 is Subset of A1 ) ) )
assume A1: f1 | (A1 /\ A2) = f2 | (A1 /\ A2) ; ::_thesis: ( ( A1 is Subset of A2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies A1 is Subset of A2 ) & ( A2 is Subset of A1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies A2 is Subset of A1 ) )
A2: now__::_thesis:_(_A1_is_Subset_of_A2_implies_f1_union_f2_=_f2_)
assume A1 is Subset of A2 ; ::_thesis: f1 union f2 = f2
then A2 = A1 \/ A2 by XBOOLE_1:12;
then (f1 union f2) | (A1 \/ A2) = f2 by A1, Def1;
then (f1 union f2) * (id (A1 \/ A2)) = f2 by RELAT_1:65;
hence f1 union f2 = f2 by FUNCT_2:17; ::_thesis: verum
end;
now__::_thesis:_(_f1_union_f2_=_f2_implies_A1_is_Subset_of_A2_)
A3: ( dom (f1 union f2) = A1 \/ A2 & dom f2 = A2 ) by FUNCT_2:def_1;
assume f1 union f2 = f2 ; ::_thesis: A1 is Subset of A2
hence A1 is Subset of A2 by A3, XBOOLE_1:7; ::_thesis: verum
end;
hence ( A1 is Subset of A2 iff f1 union f2 = f2 ) by A2; ::_thesis: ( A2 is Subset of A1 iff f1 union f2 = f1 )
A4: now__::_thesis:_(_A2_is_Subset_of_A1_implies_f1_union_f2_=_f1_)
assume A2 is Subset of A1 ; ::_thesis: f1 union f2 = f1
then A1 = A1 \/ A2 by XBOOLE_1:12;
then (f1 union f2) | (A1 \/ A2) = f1 by A1, Def1;
then (f1 union f2) * (id (A1 \/ A2)) = f1 by RELAT_1:65;
hence f1 union f2 = f1 by FUNCT_2:17; ::_thesis: verum
end;
now__::_thesis:_(_f1_union_f2_=_f1_implies_A2_is_Subset_of_A1_)
A5: ( dom (f1 union f2) = A1 \/ A2 & dom f1 = A1 ) by FUNCT_2:def_1;
assume f1 union f2 = f1 ; ::_thesis: A2 is Subset of A1
hence A2 is Subset of A1 by A5, XBOOLE_1:7; ::_thesis: verum
end;
hence ( A2 is Subset of A1 iff f1 union f2 = f1 ) by A4; ::_thesis: verum
end;
begin
theorem Th6: :: TMAP_1:6
for X being TopStruct
for X0 being SubSpace of X holds TopStruct(# the carrier of X0, the topology of X0 #) is strict SubSpace of X
proof
let X be TopStruct ; ::_thesis: for X0 being SubSpace of X holds TopStruct(# the carrier of X0, the topology of X0 #) is strict SubSpace of X
let X0 be SubSpace of X; ::_thesis: TopStruct(# the carrier of X0, the topology of X0 #) is strict SubSpace of X
reconsider S = TopStruct(# the carrier of X0, the topology of X0 #) as TopStruct ;
S is SubSpace of X
proof
A1: [#] X0 = the carrier of X0 ;
hence [#] S c= [#] X by PRE_TOPC:def_4; :: according to PRE_TOPC:def_4 ::_thesis: for b1 being Element of bool the carrier of S holds
( ( not b1 in the topology of S or ex b2 being Element of bool the carrier of X st
( b2 in the topology of X & b1 = b2 /\ ([#] S) ) ) & ( for b2 being Element of bool the carrier of X holds
( not b2 in the topology of X or not b1 = b2 /\ ([#] S) ) or b1 in the topology of S ) )
let P be Subset of S; ::_thesis: ( ( not P in the topology of S or ex b1 being Element of bool the carrier of X st
( b1 in the topology of X & P = b1 /\ ([#] S) ) ) & ( for b1 being Element of bool the carrier of X holds
( not b1 in the topology of X or not P = b1 /\ ([#] S) ) or P in the topology of S ) )
thus ( P in the topology of S implies ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] S) ) ) by A1, PRE_TOPC:def_4; ::_thesis: ( for b1 being Element of bool the carrier of X holds
( not b1 in the topology of X or not P = b1 /\ ([#] S) ) or P in the topology of S )
given Q being Subset of X such that A2: ( Q in the topology of X & P = Q /\ ([#] S) ) ; ::_thesis: P in the topology of S
thus P in the topology of S by A1, A2, PRE_TOPC:def_4; ::_thesis: verum
end;
hence TopStruct(# the carrier of X0, the topology of X0 #) is strict SubSpace of X ; ::_thesis: verum
end;
theorem Th7: :: TMAP_1:7
for X being TopStruct
for X1, X2 being TopSpace st X1 = TopStruct(# the carrier of X2, the topology of X2 #) holds
( X1 is SubSpace of X iff X2 is SubSpace of X )
proof
let X be TopStruct ; ::_thesis: for X1, X2 being TopSpace st X1 = TopStruct(# the carrier of X2, the topology of X2 #) holds
( X1 is SubSpace of X iff X2 is SubSpace of X )
let X1, X2 be TopSpace; ::_thesis: ( X1 = TopStruct(# the carrier of X2, the topology of X2 #) implies ( X1 is SubSpace of X iff X2 is SubSpace of X ) )
assume A1: X1 = TopStruct(# the carrier of X2, the topology of X2 #) ; ::_thesis: ( X1 is SubSpace of X iff X2 is SubSpace of X )
thus ( X1 is SubSpace of X implies X2 is SubSpace of X ) ::_thesis: ( X2 is SubSpace of X implies X1 is SubSpace of X )
proof
A2: [#] X1 = the carrier of X1 ;
assume A3: X1 is SubSpace of X ; ::_thesis: X2 is SubSpace of X
hence [#] X2 c= [#] X by A1, A2, PRE_TOPC:def_4; :: according to PRE_TOPC:def_4 ::_thesis: for b1 being Element of bool the carrier of X2 holds
( ( not b1 in the topology of X2 or ex b2 being Element of bool the carrier of X st
( b2 in the topology of X & b1 = b2 /\ ([#] X2) ) ) & ( for b2 being Element of bool the carrier of X holds
( not b2 in the topology of X or not b1 = b2 /\ ([#] X2) ) or b1 in the topology of X2 ) )
let P be Subset of X2; ::_thesis: ( ( not P in the topology of X2 or ex b1 being Element of bool the carrier of X st
( b1 in the topology of X & P = b1 /\ ([#] X2) ) ) & ( for b1 being Element of bool the carrier of X holds
( not b1 in the topology of X or not P = b1 /\ ([#] X2) ) or P in the topology of X2 ) )
thus ( P in the topology of X2 implies ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X2) ) ) by A1, A3, A2, PRE_TOPC:def_4; ::_thesis: ( for b1 being Element of bool the carrier of X holds
( not b1 in the topology of X or not P = b1 /\ ([#] X2) ) or P in the topology of X2 )
given Q being Subset of X such that A4: ( Q in the topology of X & P = Q /\ ([#] X2) ) ; ::_thesis: P in the topology of X2
thus P in the topology of X2 by A1, A3, A2, A4, PRE_TOPC:def_4; ::_thesis: verum
end;
thus ( X2 is SubSpace of X implies X1 is SubSpace of X ) by A1, Th6; ::_thesis: verum
end;
theorem Th8: :: TMAP_1:8
for X, X1, X2 being TopSpace st X2 = TopStruct(# the carrier of X1, the topology of X1 #) holds
( X1 is closed SubSpace of X iff X2 is closed SubSpace of X )
proof
let X be TopSpace; ::_thesis: for X1, X2 being TopSpace st X2 = TopStruct(# the carrier of X1, the topology of X1 #) holds
( X1 is closed SubSpace of X iff X2 is closed SubSpace of X )
let X1, X2 be TopSpace; ::_thesis: ( X2 = TopStruct(# the carrier of X1, the topology of X1 #) implies ( X1 is closed SubSpace of X iff X2 is closed SubSpace of X ) )
assume A1: X2 = TopStruct(# the carrier of X1, the topology of X1 #) ; ::_thesis: ( X1 is closed SubSpace of X iff X2 is closed SubSpace of X )
thus ( X1 is closed SubSpace of X implies X2 is closed SubSpace of X ) ::_thesis: ( X2 is closed SubSpace of X implies X1 is closed SubSpace of X )
proof
assume A2: X1 is closed SubSpace of X ; ::_thesis: X2 is closed SubSpace of X
then reconsider Y2 = X2 as SubSpace of X by A1, Th7;
reconsider A2 = the carrier of Y2 as Subset of X by TSEP_1:1;
A2 is closed by A1, A2, TSEP_1:11;
hence X2 is closed SubSpace of X by TSEP_1:11; ::_thesis: verum
end;
assume A3: X2 is closed SubSpace of X ; ::_thesis: X1 is closed SubSpace of X
then reconsider Y1 = X1 as SubSpace of X by A1, Th7;
reconsider A1 = the carrier of Y1 as Subset of X by TSEP_1:1;
A1 is closed by A1, A3, TSEP_1:11;
hence X1 is closed SubSpace of X by TSEP_1:11; ::_thesis: verum
end;
theorem Th9: :: TMAP_1:9
for X, X1, X2 being TopSpace st X2 = TopStruct(# the carrier of X1, the topology of X1 #) holds
( X1 is open SubSpace of X iff X2 is open SubSpace of X )
proof
let X be TopSpace; ::_thesis: for X1, X2 being TopSpace st X2 = TopStruct(# the carrier of X1, the topology of X1 #) holds
( X1 is open SubSpace of X iff X2 is open SubSpace of X )
let X1, X2 be TopSpace; ::_thesis: ( X2 = TopStruct(# the carrier of X1, the topology of X1 #) implies ( X1 is open SubSpace of X iff X2 is open SubSpace of X ) )
assume A1: X2 = TopStruct(# the carrier of X1, the topology of X1 #) ; ::_thesis: ( X1 is open SubSpace of X iff X2 is open SubSpace of X )
thus ( X1 is open SubSpace of X implies X2 is open SubSpace of X ) ::_thesis: ( X2 is open SubSpace of X implies X1 is open SubSpace of X )
proof
assume A2: X1 is open SubSpace of X ; ::_thesis: X2 is open SubSpace of X
then reconsider Y2 = X2 as SubSpace of X by A1, Th7;
reconsider A2 = the carrier of Y2 as Subset of X by TSEP_1:1;
A2 is open by A1, A2, TSEP_1:16;
hence X2 is open SubSpace of X by TSEP_1:16; ::_thesis: verum
end;
assume A3: X2 is open SubSpace of X ; ::_thesis: X1 is open SubSpace of X
then reconsider Y1 = X1 as SubSpace of X by A1, Th7;
reconsider A1 = the carrier of Y1 as Subset of X by TSEP_1:1;
A1 is open by A1, A3, TSEP_1:16;
hence X1 is open SubSpace of X by TSEP_1:16; ::_thesis: verum
end;
theorem Th10: :: TMAP_1:10
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 is SubSpace of X2 holds
for x1 being Point of X1 ex x2 being Point of X2 st x2 = x1
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 is SubSpace of X2 holds
for x1 being Point of X1 ex x2 being Point of X2 st x2 = x1
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 is SubSpace of X2 implies for x1 being Point of X1 ex x2 being Point of X2 st x2 = x1 )
assume X1 is SubSpace of X2 ; ::_thesis: for x1 being Point of X1 ex x2 being Point of X2 st x2 = x1
then A1: the carrier of X1 c= the carrier of X2 by TSEP_1:4;
let x1 be Point of X1; ::_thesis: ex x2 being Point of X2 st x2 = x1
x1 in the carrier of X1 ;
then reconsider x2 = x1 as Point of X2 by A1;
take x2 ; ::_thesis: x2 = x1
thus x2 = x1 ; ::_thesis: verum
end;
theorem Th11: :: TMAP_1:11
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2) holds
( ex x1 being Point of X1 st x1 = x or ex x2 being Point of X2 st x2 = x )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2) holds
( ex x1 being Point of X1 st x1 = x or ex x2 being Point of X2 st x2 = x )
let X1, X2 be non empty SubSpace of X; ::_thesis: for x being Point of (X1 union X2) holds
( ex x1 being Point of X1 st x1 = x or ex x2 being Point of X2 st x2 = x )
let x be Point of (X1 union X2); ::_thesis: ( ex x1 being Point of X1 st x1 = x or ex x2 being Point of X2 st x2 = x )
reconsider A0 = the carrier of (X1 union X2) as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume A1: for x1 being Point of X1 holds not x1 = x ; ::_thesis: ex x2 being Point of X2 st x2 = x
ex x2 being Point of X2 st x2 = x
proof
( A0 = A1 \/ A2 & not x in A1 ) by A1, TSEP_1:def_2;
then reconsider x2 = x as Point of X2 by XBOOLE_0:def_3;
take x2 ; ::_thesis: x2 = x
thus x2 = x ; ::_thesis: verum
end;
hence ex x2 being Point of X2 st x2 = x ; ::_thesis: verum
end;
theorem Th12: :: TMAP_1:12
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
for x being Point of (X1 meet X2) holds
( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
for x being Point of (X1 meet X2) holds
( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies for x being Point of (X1 meet X2) holds
( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x ) )
assume A1: X1 meets X2 ; ::_thesis: for x being Point of (X1 meet X2) holds
( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x )
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
reconsider A0 = the carrier of (X1 meet X2) as Subset of X by TSEP_1:1;
let x be Point of (X1 meet X2); ::_thesis: ( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x )
A0 = A1 /\ A2 by A1, TSEP_1:def_4;
then ( x in A1 & x in A2 ) by XBOOLE_0:def_4;
hence ( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x ) ; ::_thesis: verum
end;
theorem :: TMAP_1:13
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for F1 being Subset of X1
for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds
ex H being Subset of (X1 union X2) st
( H is closed & x in H & H c= F1 \/ F2 )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for F1 being Subset of X1
for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds
ex H being Subset of (X1 union X2) st
( H is closed & x in H & H c= F1 \/ F2 )
let X1, X2 be non empty SubSpace of X; ::_thesis: for x being Point of (X1 union X2)
for F1 being Subset of X1
for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds
ex H being Subset of (X1 union X2) st
( H is closed & x in H & H c= F1 \/ F2 )
let x be Point of (X1 union X2); ::_thesis: for F1 being Subset of X1
for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds
ex H being Subset of (X1 union X2) st
( H is closed & x in H & H c= F1 \/ F2 )
let F1 be Subset of X1; ::_thesis: for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds
ex H being Subset of (X1 union X2) st
( H is closed & x in H & H c= F1 \/ F2 )
let F2 be Subset of X2; ::_thesis: ( F1 is closed & x in F1 & F2 is closed & x in F2 implies ex H being Subset of (X1 union X2) st
( H is closed & x in H & H c= F1 \/ F2 ) )
assume that
A1: F1 is closed and
A2: x in F1 and
A3: F2 is closed and
A4: x in F2 ; ::_thesis: ex H being Subset of (X1 union X2) st
( H is closed & x in H & H c= F1 \/ F2 )
A5: X1 is SubSpace of X1 union X2 by TSEP_1:22;
then reconsider C1 = the carrier of X1 as Subset of (X1 union X2) by TSEP_1:1;
consider H1 being Subset of (X1 union X2) such that
A6: H1 is closed and
A7: H1 /\ ([#] X1) = F1 by A1, A5, PRE_TOPC:13;
A8: x in H1 by A2, A7, XBOOLE_0:def_4;
A9: X2 is SubSpace of X1 union X2 by TSEP_1:22;
then reconsider C2 = the carrier of X2 as Subset of (X1 union X2) by TSEP_1:1;
consider H2 being Subset of (X1 union X2) such that
A10: H2 is closed and
A11: H2 /\ ([#] X2) = F2 by A3, A9, PRE_TOPC:13;
A12: x in H2 by A4, A11, XBOOLE_0:def_4;
take H = H1 /\ H2; ::_thesis: ( H is closed & x in H & H c= F1 \/ F2 )
A13: ( H /\ C1 c= H1 /\ C1 & H /\ C2 c= H2 /\ C2 ) by XBOOLE_1:17, XBOOLE_1:26;
the carrier of (X1 union X2) = C1 \/ C2 by TSEP_1:def_2;
then H = H /\ (C1 \/ C2) by XBOOLE_1:28
.= (H /\ C1) \/ (H /\ C2) by XBOOLE_1:23 ;
hence ( H is closed & x in H & H c= F1 \/ F2 ) by A6, A7, A10, A11, A13, A8, A12, XBOOLE_0:def_4, XBOOLE_1:13; ::_thesis: verum
end;
theorem Th14: :: TMAP_1:14
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for U1 being Subset of X1
for U2 being Subset of X2 st U1 is open & x in U1 & U2 is open & x in U2 holds
ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for U1 being Subset of X1
for U2 being Subset of X2 st U1 is open & x in U1 & U2 is open & x in U2 holds
ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
let X1, X2 be non empty SubSpace of X; ::_thesis: for x being Point of (X1 union X2)
for U1 being Subset of X1
for U2 being Subset of X2 st U1 is open & x in U1 & U2 is open & x in U2 holds
ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
let x be Point of (X1 union X2); ::_thesis: for U1 being Subset of X1
for U2 being Subset of X2 st U1 is open & x in U1 & U2 is open & x in U2 holds
ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
let U1 be Subset of X1; ::_thesis: for U2 being Subset of X2 st U1 is open & x in U1 & U2 is open & x in U2 holds
ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
let U2 be Subset of X2; ::_thesis: ( U1 is open & x in U1 & U2 is open & x in U2 implies ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 ) )
assume that
A1: U1 is open and
A2: x in U1 and
A3: U2 is open and
A4: x in U2 ; ::_thesis: ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
A5: X1 is SubSpace of X1 union X2 by TSEP_1:22;
then reconsider C1 = the carrier of X1 as Subset of (X1 union X2) by TSEP_1:1;
consider V1 being Subset of (X1 union X2) such that
A6: V1 is open and
A7: V1 /\ ([#] X1) = U1 by A1, A5, TOPS_2:24;
A8: x in V1 by A2, A7, XBOOLE_0:def_4;
A9: X2 is SubSpace of X1 union X2 by TSEP_1:22;
then reconsider C2 = the carrier of X2 as Subset of (X1 union X2) by TSEP_1:1;
consider V2 being Subset of (X1 union X2) such that
A10: V2 is open and
A11: V2 /\ ([#] X2) = U2 by A3, A9, TOPS_2:24;
A12: x in V2 by A4, A11, XBOOLE_0:def_4;
take V = V1 /\ V2; ::_thesis: ( V is open & x in V & V c= U1 \/ U2 )
A13: ( V /\ C1 c= V1 /\ C1 & V /\ C2 c= V2 /\ C2 ) by XBOOLE_1:17, XBOOLE_1:26;
the carrier of (X1 union X2) = C1 \/ C2 by TSEP_1:def_2;
then V = V /\ (C1 \/ C2) by XBOOLE_1:28
.= (V /\ C1) \/ (V /\ C2) by XBOOLE_1:23 ;
hence ( V is open & x in V & V c= U1 \/ U2 ) by A6, A7, A10, A11, A13, A8, A12, XBOOLE_0:def_4, XBOOLE_1:13; ::_thesis: verum
end;
theorem Th15: :: TMAP_1:15
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for x1 being Point of X1
for x2 being Point of X2 st x1 = x & x2 = x holds
for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for x1 being Point of X1
for x2 being Point of X2 st x1 = x & x2 = x holds
for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
let X1, X2 be non empty SubSpace of X; ::_thesis: for x being Point of (X1 union X2)
for x1 being Point of X1
for x2 being Point of X2 st x1 = x & x2 = x holds
for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
let x be Point of (X1 union X2); ::_thesis: for x1 being Point of X1
for x2 being Point of X2 st x1 = x & x2 = x holds
for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
let x1 be Point of X1; ::_thesis: for x2 being Point of X2 st x1 = x & x2 = x holds
for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
let x2 be Point of X2; ::_thesis: ( x1 = x & x2 = x implies for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 ) )
assume A1: ( x1 = x & x2 = x ) ; ::_thesis: for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
let A1 be a_neighborhood of x1; ::_thesis: for A2 being a_neighborhood of x2 ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
let A2 be a_neighborhood of x2; ::_thesis: ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
consider U1 being Subset of X1 such that
A2: U1 is open and
A3: U1 c= A1 and
A4: x1 in U1 by CONNSP_2:6;
consider U2 being Subset of X2 such that
A5: U2 is open and
A6: U2 c= A2 and
A7: x2 in U2 by CONNSP_2:6;
consider V being Subset of (X1 union X2) such that
A8: ( V is open & x in V & V c= U1 \/ U2 ) by A1, A2, A4, A5, A7, Th14;
take V ; ::_thesis: ( V is open & x in V & V c= A1 \/ A2 )
U1 \/ U2 c= A1 \/ A2 by A3, A6, XBOOLE_1:13;
hence ( V is open & x in V & V c= A1 \/ A2 ) by A8, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th16: :: TMAP_1:16
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for x1 being Point of X1
for x2 being Point of X2 st x1 = x & x2 = x holds
for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for x1 being Point of X1
for x2 being Point of X2 st x1 = x & x2 = x holds
for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2
let X1, X2 be non empty SubSpace of X; ::_thesis: for x being Point of (X1 union X2)
for x1 being Point of X1
for x2 being Point of X2 st x1 = x & x2 = x holds
for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2
let x be Point of (X1 union X2); ::_thesis: for x1 being Point of X1
for x2 being Point of X2 st x1 = x & x2 = x holds
for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2
let x1 be Point of X1; ::_thesis: for x2 being Point of X2 st x1 = x & x2 = x holds
for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2
let x2 be Point of X2; ::_thesis: ( x1 = x & x2 = x implies for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2 )
assume A1: ( x1 = x & x2 = x ) ; ::_thesis: for A1 being a_neighborhood of x1
for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2
let A1 be a_neighborhood of x1; ::_thesis: for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2
let A2 be a_neighborhood of x2; ::_thesis: ex A being a_neighborhood of x st A c= A1 \/ A2
consider V being Subset of (X1 union X2) such that
A2: ( V is open & x in V ) and
A3: V c= A1 \/ A2 by A1, Th15;
reconsider W = V as a_neighborhood of x by A2, CONNSP_2:3;
take W ; ::_thesis: W c= A1 \/ A2
thus W c= A1 \/ A2 by A3; ::_thesis: verum
end;
theorem Th17: :: TMAP_1:17
for X being non empty TopSpace
for X0, X1 being non empty SubSpace of X st X0 is SubSpace of X1 holds
( X0 meets X1 & X1 meets X0 )
proof
let X be non empty TopSpace; ::_thesis: for X0, X1 being non empty SubSpace of X st X0 is SubSpace of X1 holds
( X0 meets X1 & X1 meets X0 )
let X0, X1 be non empty SubSpace of X; ::_thesis: ( X0 is SubSpace of X1 implies ( X0 meets X1 & X1 meets X0 ) )
assume X0 is SubSpace of X1 ; ::_thesis: ( X0 meets X1 & X1 meets X0 )
then the carrier of X0 c= the carrier of X1 by TSEP_1:4;
then the carrier of X0 = the carrier of X0 /\ the carrier of X1 by XBOOLE_1:28;
then A1: the carrier of X0 meets the carrier of X1 by XBOOLE_0:def_7;
hence X0 meets X1 by TSEP_1:def_3; ::_thesis: X1 meets X0
thus X1 meets X0 by A1, TSEP_1:def_3; ::_thesis: verum
end;
theorem Th18: :: TMAP_1:18
for X being non empty TopSpace
for X0, X1, X2 being non empty SubSpace of X st X0 is SubSpace of X1 & ( X0 meets X2 or X2 meets X0 ) holds
( X1 meets X2 & X2 meets X1 )
proof
let X be non empty TopSpace; ::_thesis: for X0, X1, X2 being non empty SubSpace of X st X0 is SubSpace of X1 & ( X0 meets X2 or X2 meets X0 ) holds
( X1 meets X2 & X2 meets X1 )
let X0, X1, X2 be non empty SubSpace of X; ::_thesis: ( X0 is SubSpace of X1 & ( X0 meets X2 or X2 meets X0 ) implies ( X1 meets X2 & X2 meets X1 ) )
reconsider A0 = the carrier of X0, A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume X0 is SubSpace of X1 ; ::_thesis: ( ( not X0 meets X2 & not X2 meets X0 ) or ( X1 meets X2 & X2 meets X1 ) )
then A1: A0 c= A1 by TSEP_1:4;
A2: now__::_thesis:_(_X0_meets_X2_&_(_X0_meets_X2_or_X2_meets_X0_)_implies_(_X1_meets_X2_&_X2_meets_X1_)_)
assume X0 meets X2 ; ::_thesis: ( ( not X0 meets X2 & not X2 meets X0 ) or ( X1 meets X2 & X2 meets X1 ) )
then A2 meets A0 by TSEP_1:def_3;
then A2 meets A1 by A1, XBOOLE_1:63;
hence ( ( not X0 meets X2 & not X2 meets X0 ) or ( X1 meets X2 & X2 meets X1 ) ) by TSEP_1:def_3; ::_thesis: verum
end;
assume ( X0 meets X2 or X2 meets X0 ) ; ::_thesis: ( X1 meets X2 & X2 meets X1 )
hence ( X1 meets X2 & X2 meets X1 ) by A2; ::_thesis: verum
end;
theorem Th19: :: TMAP_1:19
for X being non empty TopSpace
for X0, X1, X2 being non empty SubSpace of X st X0 is SubSpace of X1 & ( X1 misses X2 or X2 misses X1 ) holds
( X0 misses X2 & X2 misses X0 )
proof
let X be non empty TopSpace; ::_thesis: for X0, X1, X2 being non empty SubSpace of X st X0 is SubSpace of X1 & ( X1 misses X2 or X2 misses X1 ) holds
( X0 misses X2 & X2 misses X0 )
let X0, X1, X2 be non empty SubSpace of X; ::_thesis: ( X0 is SubSpace of X1 & ( X1 misses X2 or X2 misses X1 ) implies ( X0 misses X2 & X2 misses X0 ) )
reconsider A0 = the carrier of X0, A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume X0 is SubSpace of X1 ; ::_thesis: ( ( not X1 misses X2 & not X2 misses X1 ) or ( X0 misses X2 & X2 misses X0 ) )
then A1: A0 c= A1 by TSEP_1:4;
A2: now__::_thesis:_(_X1_misses_X2_&_(_X1_misses_X2_or_X2_misses_X1_)_implies_(_X0_misses_X2_&_X2_misses_X0_)_)
assume X1 misses X2 ; ::_thesis: ( ( not X1 misses X2 & not X2 misses X1 ) or ( X0 misses X2 & X2 misses X0 ) )
then A2 misses A1 by TSEP_1:def_3;
then A2 misses A0 by A1, XBOOLE_1:63;
hence ( ( not X1 misses X2 & not X2 misses X1 ) or ( X0 misses X2 & X2 misses X0 ) ) by TSEP_1:def_3; ::_thesis: verum
end;
assume ( X1 misses X2 or X2 misses X1 ) ; ::_thesis: ( X0 misses X2 & X2 misses X0 )
hence ( X0 misses X2 & X2 misses X0 ) by A2; ::_thesis: verum
end;
theorem :: TMAP_1:20
for X being non empty TopSpace
for X0 being non empty SubSpace of X holds X0 union X0 = TopStruct(# the carrier of X0, the topology of X0 #)
proof
let X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X holds X0 union X0 = TopStruct(# the carrier of X0, the topology of X0 #)
let X0 be non empty SubSpace of X; ::_thesis: X0 union X0 = TopStruct(# the carrier of X0, the topology of X0 #)
X0 is SubSpace of X0 by TSEP_1:2;
hence X0 union X0 = TopStruct(# the carrier of X0, the topology of X0 #) by TSEP_1:23; ::_thesis: verum
end;
theorem :: TMAP_1:21
for X being non empty TopSpace
for X0 being non empty SubSpace of X holds X0 meet X0 = TopStruct(# the carrier of X0, the topology of X0 #)
proof
let X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X holds X0 meet X0 = TopStruct(# the carrier of X0, the topology of X0 #)
let X0 be non empty SubSpace of X; ::_thesis: X0 meet X0 = TopStruct(# the carrier of X0, the topology of X0 #)
A1: X0 is SubSpace of X0 by TSEP_1:2;
then X0 meets X0 by Th17;
hence X0 meet X0 = TopStruct(# the carrier of X0, the topology of X0 #) by A1, TSEP_1:28; ::_thesis: verum
end;
theorem Th22: :: TMAP_1:22
for X being non empty TopSpace
for Y1, X1, Y2, X2 being non empty SubSpace of X st Y1 is SubSpace of X1 & Y2 is SubSpace of X2 holds
Y1 union Y2 is SubSpace of X1 union X2
proof
let X be non empty TopSpace; ::_thesis: for Y1, X1, Y2, X2 being non empty SubSpace of X st Y1 is SubSpace of X1 & Y2 is SubSpace of X2 holds
Y1 union Y2 is SubSpace of X1 union X2
let Y1, X1, Y2, X2 be non empty SubSpace of X; ::_thesis: ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 implies Y1 union Y2 is SubSpace of X1 union X2 )
assume ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 ) ; ::_thesis: Y1 union Y2 is SubSpace of X1 union X2
then ( the carrier of Y1 c= the carrier of X1 & the carrier of Y2 c= the carrier of X2 ) by TSEP_1:4;
then the carrier of Y1 \/ the carrier of Y2 c= the carrier of X1 \/ the carrier of X2 by XBOOLE_1:13;
then the carrier of (Y1 union Y2) c= the carrier of X1 \/ the carrier of X2 by TSEP_1:def_2;
then the carrier of (Y1 union Y2) c= the carrier of (X1 union X2) by TSEP_1:def_2;
hence Y1 union Y2 is SubSpace of X1 union X2 by TSEP_1:4; ::_thesis: verum
end;
theorem :: TMAP_1:23
for X being non empty TopSpace
for Y1, Y2, X1, X2 being non empty SubSpace of X st Y1 meets Y2 & Y1 is SubSpace of X1 & Y2 is SubSpace of X2 holds
Y1 meet Y2 is SubSpace of X1 meet X2
proof
let X be non empty TopSpace; ::_thesis: for Y1, Y2, X1, X2 being non empty SubSpace of X st Y1 meets Y2 & Y1 is SubSpace of X1 & Y2 is SubSpace of X2 holds
Y1 meet Y2 is SubSpace of X1 meet X2
let Y1, Y2, X1, X2 be non empty SubSpace of X; ::_thesis: ( Y1 meets Y2 & Y1 is SubSpace of X1 & Y2 is SubSpace of X2 implies Y1 meet Y2 is SubSpace of X1 meet X2 )
assume A1: Y1 meets Y2 ; ::_thesis: ( not Y1 is SubSpace of X1 or not Y2 is SubSpace of X2 or Y1 meet Y2 is SubSpace of X1 meet X2 )
assume ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 ) ; ::_thesis: Y1 meet Y2 is SubSpace of X1 meet X2
then A2: ( the carrier of Y1 c= the carrier of X1 & the carrier of Y2 c= the carrier of X2 ) by TSEP_1:4;
the carrier of Y1 meets the carrier of Y2 by A1, TSEP_1:def_3;
then the carrier of Y1 /\ the carrier of Y2 <> {} by XBOOLE_0:def_7;
then the carrier of X1 /\ the carrier of X2 <> {} by A2, XBOOLE_1:3, XBOOLE_1:27;
then the carrier of X1 meets the carrier of X2 by XBOOLE_0:def_7;
then A3: X1 meets X2 by TSEP_1:def_3;
the carrier of Y1 /\ the carrier of Y2 c= the carrier of X1 /\ the carrier of X2 by A2, XBOOLE_1:27;
then the carrier of Y1 /\ the carrier of Y2 c= the carrier of (X1 meet X2) by A3, TSEP_1:def_4;
then the carrier of (Y1 meet Y2) c= the carrier of (X1 meet X2) by A1, TSEP_1:def_4;
hence Y1 meet Y2 is SubSpace of X1 meet X2 by TSEP_1:4; ::_thesis: verum
end;
theorem Th24: :: TMAP_1:24
for X being non empty TopSpace
for X1, X0, X2 being non empty SubSpace of X st X1 is SubSpace of X0 & X2 is SubSpace of X0 holds
X1 union X2 is SubSpace of X0
proof
let X be non empty TopSpace; ::_thesis: for X1, X0, X2 being non empty SubSpace of X st X1 is SubSpace of X0 & X2 is SubSpace of X0 holds
X1 union X2 is SubSpace of X0
let X1, X0, X2 be non empty SubSpace of X; ::_thesis: ( X1 is SubSpace of X0 & X2 is SubSpace of X0 implies X1 union X2 is SubSpace of X0 )
assume ( X1 is SubSpace of X0 & X2 is SubSpace of X0 ) ; ::_thesis: X1 union X2 is SubSpace of X0
then ( the carrier of X1 c= the carrier of X0 & the carrier of X2 c= the carrier of X0 ) by TSEP_1:4;
then the carrier of X1 \/ the carrier of X2 c= the carrier of X0 by XBOOLE_1:8;
then the carrier of (X1 union X2) c= the carrier of X0 by TSEP_1:def_2;
hence X1 union X2 is SubSpace of X0 by TSEP_1:4; ::_thesis: verum
end;
theorem :: TMAP_1:25
for X being non empty TopSpace
for X1, X2, X0 being non empty SubSpace of X st X1 meets X2 & X1 is SubSpace of X0 & X2 is SubSpace of X0 holds
X1 meet X2 is SubSpace of X0
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, X0 being non empty SubSpace of X st X1 meets X2 & X1 is SubSpace of X0 & X2 is SubSpace of X0 holds
X1 meet X2 is SubSpace of X0
let X1, X2, X0 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 & X1 is SubSpace of X0 & X2 is SubSpace of X0 implies X1 meet X2 is SubSpace of X0 )
assume A1: X1 meets X2 ; ::_thesis: ( not X1 is SubSpace of X0 or not X2 is SubSpace of X0 or X1 meet X2 is SubSpace of X0 )
assume ( X1 is SubSpace of X0 & X2 is SubSpace of X0 ) ; ::_thesis: X1 meet X2 is SubSpace of X0
then ( the carrier of X1 c= the carrier of X0 & the carrier of X2 c= the carrier of X0 ) by TSEP_1:4;
then A2: the carrier of X1 \/ the carrier of X2 c= the carrier of X0 by XBOOLE_1:8;
the carrier of X1 /\ the carrier of X2 c= the carrier of X1 \/ the carrier of X2 by XBOOLE_1:29;
then the carrier of X1 /\ the carrier of X2 c= the carrier of X0 by A2, XBOOLE_1:1;
then the carrier of (X1 meet X2) c= the carrier of X0 by A1, TSEP_1:def_4;
hence X1 meet X2 is SubSpace of X0 by TSEP_1:4; ::_thesis: verum
end;
theorem Th26: :: TMAP_1:26
for X being non empty TopSpace
for X1, X0, X2 being non empty SubSpace of X holds
( ( ( X1 misses X0 or X0 misses X1 ) & ( X2 meets X0 or X0 meets X2 ) implies ( (X1 union X2) meet X0 = X2 meet X0 & X0 meet (X1 union X2) = X0 meet X2 ) ) & ( ( X1 meets X0 or X0 meets X1 ) & ( X2 misses X0 or X0 misses X2 ) implies ( (X1 union X2) meet X0 = X1 meet X0 & X0 meet (X1 union X2) = X0 meet X1 ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X0, X2 being non empty SubSpace of X holds
( ( ( X1 misses X0 or X0 misses X1 ) & ( X2 meets X0 or X0 meets X2 ) implies ( (X1 union X2) meet X0 = X2 meet X0 & X0 meet (X1 union X2) = X0 meet X2 ) ) & ( ( X1 meets X0 or X0 meets X1 ) & ( X2 misses X0 or X0 misses X2 ) implies ( (X1 union X2) meet X0 = X1 meet X0 & X0 meet (X1 union X2) = X0 meet X1 ) ) )
let X1, X0, X2 be non empty SubSpace of X; ::_thesis: ( ( ( X1 misses X0 or X0 misses X1 ) & ( X2 meets X0 or X0 meets X2 ) implies ( (X1 union X2) meet X0 = X2 meet X0 & X0 meet (X1 union X2) = X0 meet X2 ) ) & ( ( X1 meets X0 or X0 meets X1 ) & ( X2 misses X0 or X0 misses X2 ) implies ( (X1 union X2) meet X0 = X1 meet X0 & X0 meet (X1 union X2) = X0 meet X1 ) ) )
reconsider A0 = the carrier of X0, A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;
thus ( ( X1 misses X0 or X0 misses X1 ) & ( X2 meets X0 or X0 meets X2 ) implies ( (X1 union X2) meet X0 = X2 meet X0 & X0 meet (X1 union X2) = X0 meet X2 ) ) ::_thesis: ( ( X1 meets X0 or X0 meets X1 ) & ( X2 misses X0 or X0 misses X2 ) implies ( (X1 union X2) meet X0 = X1 meet X0 & X0 meet (X1 union X2) = X0 meet X1 ) )
proof
assume that
A1: ( X1 misses X0 or X0 misses X1 ) and
A2: ( X2 meets X0 or X0 meets X2 ) ; ::_thesis: ( (X1 union X2) meet X0 = X2 meet X0 & X0 meet (X1 union X2) = X0 meet X2 )
A3: A1 misses A0 by A1, TSEP_1:def_3;
X2 is SubSpace of X1 union X2 by TSEP_1:22;
then A4: X1 union X2 meets X0 by A2, Th18;
then A5: the carrier of (X0 meet (X1 union X2)) = A0 /\ the carrier of (X1 union X2) by TSEP_1:def_4
.= A0 /\ (A1 \/ A2) by TSEP_1:def_2
.= (A0 /\ A1) \/ (A0 /\ A2) by XBOOLE_1:23
.= {} \/ (A0 /\ A2) by A3, XBOOLE_0:def_7
.= the carrier of (X0 meet X2) by A2, TSEP_1:def_4 ;
the carrier of ((X1 union X2) meet X0) = the carrier of (X1 union X2) /\ A0 by A4, TSEP_1:def_4
.= (A1 \/ A2) /\ A0 by TSEP_1:def_2
.= (A1 /\ A0) \/ (A2 /\ A0) by XBOOLE_1:23
.= {} \/ (A2 /\ A0) by A3, XBOOLE_0:def_7
.= the carrier of (X2 meet X0) by A2, TSEP_1:def_4 ;
hence ( (X1 union X2) meet X0 = X2 meet X0 & X0 meet (X1 union X2) = X0 meet X2 ) by A5, TSEP_1:5; ::_thesis: verum
end;
thus ( ( X1 meets X0 or X0 meets X1 ) & ( X2 misses X0 or X0 misses X2 ) implies ( (X1 union X2) meet X0 = X1 meet X0 & X0 meet (X1 union X2) = X0 meet X1 ) ) ::_thesis: verum
proof
assume that
A6: ( X1 meets X0 or X0 meets X1 ) and
A7: ( X2 misses X0 or X0 misses X2 ) ; ::_thesis: ( (X1 union X2) meet X0 = X1 meet X0 & X0 meet (X1 union X2) = X0 meet X1 )
A8: A2 misses A0 by A7, TSEP_1:def_3;
X1 is SubSpace of X1 union X2 by TSEP_1:22;
then A9: X1 union X2 meets X0 by A6, Th18;
then A10: the carrier of (X0 meet (X1 union X2)) = A0 /\ the carrier of (X1 union X2) by TSEP_1:def_4
.= A0 /\ (A1 \/ A2) by TSEP_1:def_2
.= (A0 /\ A1) \/ (A0 /\ A2) by XBOOLE_1:23
.= (A0 /\ A1) \/ {} by A8, XBOOLE_0:def_7
.= the carrier of (X0 meet X1) by A6, TSEP_1:def_4 ;
the carrier of ((X1 union X2) meet X0) = the carrier of (X1 union X2) /\ A0 by A9, TSEP_1:def_4
.= (A1 \/ A2) /\ A0 by TSEP_1:def_2
.= (A1 /\ A0) \/ (A2 /\ A0) by XBOOLE_1:23
.= (A1 /\ A0) \/ {} by A8, XBOOLE_0:def_7
.= the carrier of (X1 meet X0) by A6, TSEP_1:def_4 ;
hence ( (X1 union X2) meet X0 = X1 meet X0 & X0 meet (X1 union X2) = X0 meet X1 ) by A10, TSEP_1:5; ::_thesis: verum
end;
end;
theorem Th27: :: TMAP_1:27
for X being non empty TopSpace
for X1, X2, X0 being non empty SubSpace of X st X1 meets X2 holds
( ( X1 is SubSpace of X0 implies X1 meet X2 is SubSpace of X0 meet X2 ) & ( X2 is SubSpace of X0 implies X1 meet X2 is SubSpace of X1 meet X0 ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, X0 being non empty SubSpace of X st X1 meets X2 holds
( ( X1 is SubSpace of X0 implies X1 meet X2 is SubSpace of X0 meet X2 ) & ( X2 is SubSpace of X0 implies X1 meet X2 is SubSpace of X1 meet X0 ) )
let X1, X2, X0 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies ( ( X1 is SubSpace of X0 implies X1 meet X2 is SubSpace of X0 meet X2 ) & ( X2 is SubSpace of X0 implies X1 meet X2 is SubSpace of X1 meet X0 ) ) )
reconsider A0 = the carrier of X0, A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume A1: X1 meets X2 ; ::_thesis: ( ( X1 is SubSpace of X0 implies X1 meet X2 is SubSpace of X0 meet X2 ) & ( X2 is SubSpace of X0 implies X1 meet X2 is SubSpace of X1 meet X0 ) )
then A2: the carrier of (X1 meet X2) = A1 /\ A2 by TSEP_1:def_4;
A3: now__::_thesis:_(_X2_is_SubSpace_of_X0_implies_X1_meet_X2_is_SubSpace_of_X1_meet_X0_)
assume A4: X2 is SubSpace of X0 ; ::_thesis: X1 meet X2 is SubSpace of X1 meet X0
then A2 c= A0 by TSEP_1:4;
then A5: A1 /\ A2 c= A1 /\ A0 by XBOOLE_1:26;
X1 meets X0 by A1, A4, Th18;
then the carrier of (X1 meet X0) = A1 /\ A0 by TSEP_1:def_4;
hence X1 meet X2 is SubSpace of X1 meet X0 by A2, A5, TSEP_1:4; ::_thesis: verum
end;
now__::_thesis:_(_X1_is_SubSpace_of_X0_implies_X1_meet_X2_is_SubSpace_of_X0_meet_X2_)
assume A6: X1 is SubSpace of X0 ; ::_thesis: X1 meet X2 is SubSpace of X0 meet X2
then A1 c= A0 by TSEP_1:4;
then A7: A1 /\ A2 c= A0 /\ A2 by XBOOLE_1:26;
X0 meets X2 by A1, A6, Th18;
then the carrier of (X0 meet X2) = A0 /\ A2 by TSEP_1:def_4;
hence X1 meet X2 is SubSpace of X0 meet X2 by A2, A7, TSEP_1:4; ::_thesis: verum
end;
hence ( ( X1 is SubSpace of X0 implies X1 meet X2 is SubSpace of X0 meet X2 ) & ( X2 is SubSpace of X0 implies X1 meet X2 is SubSpace of X1 meet X0 ) ) by A3; ::_thesis: verum
end;
theorem Th28: :: TMAP_1:28
for X being non empty TopSpace
for X1, X0, X2 being non empty SubSpace of X st X1 is SubSpace of X0 & ( X0 misses X2 or X2 misses X0 ) holds
( X0 meet (X1 union X2) = TopStruct(# the carrier of X1, the topology of X1 #) & X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X0, X2 being non empty SubSpace of X st X1 is SubSpace of X0 & ( X0 misses X2 or X2 misses X0 ) holds
( X0 meet (X1 union X2) = TopStruct(# the carrier of X1, the topology of X1 #) & X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #) )
let X1, X0, X2 be non empty SubSpace of X; ::_thesis: ( X1 is SubSpace of X0 & ( X0 misses X2 or X2 misses X0 ) implies ( X0 meet (X1 union X2) = TopStruct(# the carrier of X1, the topology of X1 #) & X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #) ) )
reconsider A0 = the carrier of X0, A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;
A1: X1 is SubSpace of X1 union X2 by TSEP_1:22;
assume A2: X1 is SubSpace of X0 ; ::_thesis: ( ( not X0 misses X2 & not X2 misses X0 ) or ( X0 meet (X1 union X2) = TopStruct(# the carrier of X1, the topology of X1 #) & X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #) ) )
then A3: A1 c= A0 by TSEP_1:4;
assume ( X0 misses X2 or X2 misses X0 ) ; ::_thesis: ( X0 meet (X1 union X2) = TopStruct(# the carrier of X1, the topology of X1 #) & X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #) )
then A4: A0 misses A2 by TSEP_1:def_3;
X0 meets X1 by A2, Th17;
then X0 meets X1 union X2 by A1, Th18;
then A5: the carrier of (X0 meet (X1 union X2)) = A0 /\ the carrier of (X1 union X2) by TSEP_1:def_4
.= A0 /\ (A1 \/ A2) by TSEP_1:def_2
.= (A0 /\ A1) \/ (A0 /\ A2) by XBOOLE_1:23
.= (A0 /\ A1) \/ {} by A4, XBOOLE_0:def_7
.= A1 by A3, XBOOLE_1:28 ;
hence X0 meet (X1 union X2) = TopStruct(# the carrier of X1, the topology of X1 #) by TSEP_1:5; ::_thesis: X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #)
thus X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #) by A5, TSEP_1:5; ::_thesis: verum
end;
theorem Th29: :: TMAP_1:29
for X being non empty TopSpace
for X1, X2, X0 being non empty SubSpace of X st X1 meets X2 holds
( ( X1 is SubSpace of X0 implies ( X0 meet X2 meets X1 & X2 meet X0 meets X1 ) ) & ( X2 is SubSpace of X0 implies ( X1 meet X0 meets X2 & X0 meet X1 meets X2 ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, X0 being non empty SubSpace of X st X1 meets X2 holds
( ( X1 is SubSpace of X0 implies ( X0 meet X2 meets X1 & X2 meet X0 meets X1 ) ) & ( X2 is SubSpace of X0 implies ( X1 meet X0 meets X2 & X0 meet X1 meets X2 ) ) )
let X1, X2, X0 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies ( ( X1 is SubSpace of X0 implies ( X0 meet X2 meets X1 & X2 meet X0 meets X1 ) ) & ( X2 is SubSpace of X0 implies ( X1 meet X0 meets X2 & X0 meet X1 meets X2 ) ) ) )
assume A1: X1 meets X2 ; ::_thesis: ( ( X1 is SubSpace of X0 implies ( X0 meet X2 meets X1 & X2 meet X0 meets X1 ) ) & ( X2 is SubSpace of X0 implies ( X1 meet X0 meets X2 & X0 meet X1 meets X2 ) ) )
A2: now__::_thesis:_(_X2_is_SubSpace_of_X0_implies_(_X1_meet_X0_meets_X2_&_X0_meet_X1_meets_X2_)_)
X1 meet X2 is SubSpace of X2 by A1, TSEP_1:27;
then A3: X1 meet X2 meets X2 by Th17;
assume A4: X2 is SubSpace of X0 ; ::_thesis: ( X1 meet X0 meets X2 & X0 meet X1 meets X2 )
then X1 meet X2 is SubSpace of X1 meet X0 by A1, Th27;
hence A5: X1 meet X0 meets X2 by A3, Th18; ::_thesis: X0 meet X1 meets X2
X1 meets X0 by A1, A4, Th18;
hence X0 meet X1 meets X2 by A5, TSEP_1:26; ::_thesis: verum
end;
now__::_thesis:_(_X1_is_SubSpace_of_X0_implies_(_X0_meet_X2_meets_X1_&_X2_meet_X0_meets_X1_)_)
X1 meet X2 is SubSpace of X1 by A1, TSEP_1:27;
then A6: X1 meet X2 meets X1 by Th17;
assume A7: X1 is SubSpace of X0 ; ::_thesis: ( X0 meet X2 meets X1 & X2 meet X0 meets X1 )
then X1 meet X2 is SubSpace of X0 meet X2 by A1, Th27;
hence A8: X0 meet X2 meets X1 by A6, Th18; ::_thesis: X2 meet X0 meets X1
X0 meets X2 by A1, A7, Th18;
hence X2 meet X0 meets X1 by A8, TSEP_1:26; ::_thesis: verum
end;
hence ( ( X1 is SubSpace of X0 implies ( X0 meet X2 meets X1 & X2 meet X0 meets X1 ) ) & ( X2 is SubSpace of X0 implies ( X1 meet X0 meets X2 & X0 meet X1 meets X2 ) ) ) by A2; ::_thesis: verum
end;
theorem Th30: :: TMAP_1:30
for X being non empty TopSpace
for X1, Y1, X2, Y2 being non empty SubSpace of X st X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) holds
( Y1 misses X2 & Y2 misses X1 )
proof
let X be non empty TopSpace; ::_thesis: for X1, Y1, X2, Y2 being non empty SubSpace of X st X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) holds
( Y1 misses X2 & Y2 misses X1 )
let X1, Y1, X2, Y2 be non empty SubSpace of X; ::_thesis: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) implies ( Y1 misses X2 & Y2 misses X1 ) )
assume that
A1: X1 is SubSpace of Y1 and
A2: X2 is SubSpace of Y2 ; ::_thesis: ( ( not Y1 misses Y2 & not Y1 meet Y2 misses X1 union X2 ) or ( Y1 misses X2 & Y2 misses X1 ) )
assume A3: ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ; ::_thesis: ( Y1 misses X2 & Y2 misses X1 )
now__::_thesis:_(_not_Y1_misses_Y2_implies_(_Y1_misses_X2_&_Y2_misses_X1_)_)
assume A4: not Y1 misses Y2 ; ::_thesis: ( Y1 misses X2 & Y2 misses X1 )
A5: now__::_thesis:_not_Y2_meets_X1
assume Y2 meets X1 ; ::_thesis: contradiction
then A6: Y1 meet Y2 meets X1 by A1, Th29;
X1 is SubSpace of X1 union X2 by TSEP_1:22;
hence contradiction by A3, A4, A6, Th18; ::_thesis: verum
end;
now__::_thesis:_not_Y1_meets_X2
assume Y1 meets X2 ; ::_thesis: contradiction
then A7: Y1 meet Y2 meets X2 by A2, Th29;
X2 is SubSpace of X1 union X2 by TSEP_1:22;
hence contradiction by A3, A4, A7, Th18; ::_thesis: verum
end;
hence ( Y1 misses X2 & Y2 misses X1 ) by A5; ::_thesis: verum
end;
hence ( Y1 misses X2 & Y2 misses X1 ) by A1, A2, Th19; ::_thesis: verum
end;
theorem Th31: :: TMAP_1:31
for X being non empty TopSpace
for X1, X2, Y1, Y2 being non empty SubSpace of X st X1 is not SubSpace of X2 & X2 is not SubSpace of X1 & X1 union X2 is SubSpace of Y1 union Y2 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 holds
( Y1 meets X1 union X2 & Y2 meets X1 union X2 )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, Y1, Y2 being non empty SubSpace of X st X1 is not SubSpace of X2 & X2 is not SubSpace of X1 & X1 union X2 is SubSpace of Y1 union Y2 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 holds
( Y1 meets X1 union X2 & Y2 meets X1 union X2 )
let X1, X2, Y1, Y2 be non empty SubSpace of X; ::_thesis: ( X1 is not SubSpace of X2 & X2 is not SubSpace of X1 & X1 union X2 is SubSpace of Y1 union Y2 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 implies ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) )
assume that
A1: X1 is not SubSpace of X2 and
A2: X2 is not SubSpace of X1 ; ::_thesis: ( not X1 union X2 is SubSpace of Y1 union Y2 or not Y1 meet (X1 union X2) is SubSpace of X1 or not Y2 meet (X1 union X2) is SubSpace of X2 or ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) )
reconsider A1 = the carrier of X1, A2 = the carrier of X2, C1 = the carrier of Y1, C2 = the carrier of Y2 as Subset of X by TSEP_1:1;
assume A3: X1 union X2 is SubSpace of Y1 union Y2 ; ::_thesis: ( not Y1 meet (X1 union X2) is SubSpace of X1 or not Y2 meet (X1 union X2) is SubSpace of X2 or ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) )
assume that
A4: Y1 meet (X1 union X2) is SubSpace of X1 and
A5: Y2 meet (X1 union X2) is SubSpace of X2 ; ::_thesis: ( Y1 meets X1 union X2 & Y2 meets X1 union X2 )
A6: the carrier of (X1 union X2) = A1 \/ A2 by TSEP_1:def_2;
A7: the carrier of (Y1 union Y2) = C1 \/ C2 by TSEP_1:def_2;
A8: now__::_thesis:_not_Y2_misses_X1_union_X2
assume Y2 misses X1 union X2 ; ::_thesis: contradiction
then A9: C2 misses A1 \/ A2 by A6, TSEP_1:def_3;
A1 \/ A2 c= C1 \/ C2 by A3, A6, A7, TSEP_1:4;
then A10: A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C1 /\ (A1 \/ A2)) \/ (C2 /\ (A1 \/ A2)) by XBOOLE_1:23
.= (C1 /\ (A1 \/ A2)) \/ {} by A9, XBOOLE_0:def_7
.= C1 /\ (A1 \/ A2) ;
then C1 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y1 meets X1 union X2 by A6, TSEP_1:def_3;
then the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A6, TSEP_1:def_4;
then A11: A1 \/ A2 c= A1 by A4, A10, TSEP_1:4;
A2 c= A1 \/ A2 by XBOOLE_1:7;
then A2 c= A1 by A11, XBOOLE_1:1;
hence contradiction by A2, TSEP_1:4; ::_thesis: verum
end;
now__::_thesis:_not_Y1_misses_X1_union_X2
assume Y1 misses X1 union X2 ; ::_thesis: contradiction
then A12: C1 misses A1 \/ A2 by A6, TSEP_1:def_3;
A1 \/ A2 c= C1 \/ C2 by A3, A6, A7, TSEP_1:4;
then A13: A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C1 /\ (A1 \/ A2)) \/ (C2 /\ (A1 \/ A2)) by XBOOLE_1:23
.= {} \/ (C2 /\ (A1 \/ A2)) by A12, XBOOLE_0:def_7
.= C2 /\ (A1 \/ A2) ;
then C2 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y2 meets X1 union X2 by A6, TSEP_1:def_3;
then the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A6, TSEP_1:def_4;
then A14: A1 \/ A2 c= A2 by A5, A13, TSEP_1:4;
A1 c= A1 \/ A2 by XBOOLE_1:7;
then A1 c= A2 by A14, XBOOLE_1:1;
hence contradiction by A1, TSEP_1:4; ::_thesis: verum
end;
hence ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) by A8; ::_thesis: verum
end;
theorem Th32: :: TMAP_1:32
for X being non empty TopSpace
for X1, X2, Y1, Y2, X0 being non empty SubSpace of X st X1 meets X2 & X1 is not SubSpace of X2 & X2 is not SubSpace of X1 & TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & X0 meet (X1 union X2) is SubSpace of X1 meet X2 holds
( Y1 meets X1 union X2 & Y2 meets X1 union X2 )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, Y1, Y2, X0 being non empty SubSpace of X st X1 meets X2 & X1 is not SubSpace of X2 & X2 is not SubSpace of X1 & TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & X0 meet (X1 union X2) is SubSpace of X1 meet X2 holds
( Y1 meets X1 union X2 & Y2 meets X1 union X2 )
let X1, X2, Y1, Y2, X0 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 & X1 is not SubSpace of X2 & X2 is not SubSpace of X1 & TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & X0 meet (X1 union X2) is SubSpace of X1 meet X2 implies ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) )
assume A1: X1 meets X2 ; ::_thesis: ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or not TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 or not Y1 meet (X1 union X2) is SubSpace of X1 or not Y2 meet (X1 union X2) is SubSpace of X2 or not X0 meet (X1 union X2) is SubSpace of X1 meet X2 or ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) )
reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider C2 = the carrier of Y2 as Subset of X by TSEP_1:1;
reconsider C1 = the carrier of Y1 as Subset of X by TSEP_1:1;
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
assume that
A2: X1 is not SubSpace of X2 and
A3: X2 is not SubSpace of X1 ; ::_thesis: ( not TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 or not Y1 meet (X1 union X2) is SubSpace of X1 or not Y2 meet (X1 union X2) is SubSpace of X2 or not X0 meet (X1 union X2) is SubSpace of X1 meet X2 or ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) )
assume A4: TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 ; ::_thesis: ( not Y1 meet (X1 union X2) is SubSpace of X1 or not Y2 meet (X1 union X2) is SubSpace of X2 or not X0 meet (X1 union X2) is SubSpace of X1 meet X2 or ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) )
assume that
A5: Y1 meet (X1 union X2) is SubSpace of X1 and
A6: Y2 meet (X1 union X2) is SubSpace of X2 ; ::_thesis: ( not X0 meet (X1 union X2) is SubSpace of X1 meet X2 or ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) )
assume A7: X0 meet (X1 union X2) is SubSpace of X1 meet X2 ; ::_thesis: ( Y1 meets X1 union X2 & Y2 meets X1 union X2 )
A8: the carrier of (X1 union X2) = A1 \/ A2 by TSEP_1:def_2;
A9: the carrier of (Y1 union Y2) = C1 \/ C2 by TSEP_1:def_2;
A10: now__::_thesis:_not_Y2_misses_X1_union_X2
assume Y2 misses X1 union X2 ; ::_thesis: contradiction
then A11: C2 misses A1 \/ A2 by A8, TSEP_1:def_3;
the carrier of X = (C1 \/ C2) \/ C by A4, A9, TSEP_1:def_2;
then A12: A1 \/ A2 = ((C2 \/ C1) \/ C) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C2 \/ (C1 \/ C)) /\ (A1 \/ A2) by XBOOLE_1:4
.= (C2 /\ (A1 \/ A2)) \/ ((C1 \/ C) /\ (A1 \/ A2)) by XBOOLE_1:23
.= {} \/ ((C1 \/ C) /\ (A1 \/ A2)) by A11, XBOOLE_0:def_7
.= (C1 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23 ;
A13: now__::_thesis:_(_C1_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A1_)
assume C1 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C1 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y1 meets X1 union X2 by A8, TSEP_1:def_3;
then A14: the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A8, TSEP_1:def_4;
then A15: C1 /\ (A1 \/ A2) c= A1 by A5, TSEP_1:4;
now__::_thesis:_A1_\/_A2_c=_A1
percases ( C /\ (A1 \/ A2) = {} or C /\ (A1 \/ A2) <> {} ) ;
suppose C /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A1
hence A1 \/ A2 c= A1 by A5, A12, A14, TSEP_1:4; ::_thesis: verum
end;
suppose C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then X0 meets X1 union X2 by A8, TSEP_1:def_3;
then A16: the carrier of (X0 meet (X1 union X2)) = C /\ (A1 \/ A2) by A8, TSEP_1:def_4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, TSEP_1:def_4;
then C /\ (A1 \/ A2) c= A1 /\ A2 by A7, A16, TSEP_1:4;
then A1 \/ A2 c= A1 \/ (A1 /\ A2) by A12, A15, XBOOLE_1:13;
hence A1 \/ A2 c= A1 by XBOOLE_1:12, XBOOLE_1:17; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A1 ; ::_thesis: verum
end;
A17: now__::_thesis:_(_C_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A1_)
assume C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then X0 meets X1 union X2 by A8, TSEP_1:def_3;
then A18: the carrier of (X0 meet (X1 union X2)) = C /\ (A1 \/ A2) by A8, TSEP_1:def_4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, TSEP_1:def_4;
then A19: C /\ (A1 \/ A2) c= A1 /\ A2 by A7, A18, TSEP_1:4;
A20: A1 /\ A2 c= A1 by XBOOLE_1:17;
then A21: C /\ (A1 \/ A2) c= A1 by A19, XBOOLE_1:1;
now__::_thesis:_A1_\/_A2_c=_A1
percases ( C1 /\ (A1 \/ A2) = {} or C1 /\ (A1 \/ A2) <> {} ) ;
suppose C1 /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A1
hence A1 \/ A2 c= A1 by A12, A19, A20, XBOOLE_1:1; ::_thesis: verum
end;
suppose C1 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C1 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y1 meets X1 union X2 by A8, TSEP_1:def_3;
then the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A8, TSEP_1:def_4;
then C1 /\ (A1 \/ A2) c= A1 by A5, TSEP_1:4;
hence A1 \/ A2 c= A1 by A12, A21, XBOOLE_1:8; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A1 ; ::_thesis: verum
end;
A2 c= A1 \/ A2 by XBOOLE_1:7;
then A2 c= A1 by A12, A13, A17, XBOOLE_1:1;
hence contradiction by A3, TSEP_1:4; ::_thesis: verum
end;
now__::_thesis:_not_Y1_misses_X1_union_X2
assume Y1 misses X1 union X2 ; ::_thesis: contradiction
then A22: C1 misses A1 \/ A2 by A8, TSEP_1:def_3;
the carrier of X = (C1 \/ C2) \/ C by A4, A9, TSEP_1:def_2;
then A23: A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C1 \/ (C2 \/ C)) /\ (A1 \/ A2) by XBOOLE_1:4
.= (C1 /\ (A1 \/ A2)) \/ ((C2 \/ C) /\ (A1 \/ A2)) by XBOOLE_1:23
.= {} \/ ((C2 \/ C) /\ (A1 \/ A2)) by A22, XBOOLE_0:def_7
.= (C2 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23 ;
A24: now__::_thesis:_(_C2_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A2_)
assume C2 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C2 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y2 meets X1 union X2 by A8, TSEP_1:def_3;
then A25: the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A8, TSEP_1:def_4;
then A26: C2 /\ (A1 \/ A2) c= A2 by A6, TSEP_1:4;
now__::_thesis:_A1_\/_A2_c=_A2
percases ( C /\ (A1 \/ A2) = {} or C /\ (A1 \/ A2) <> {} ) ;
suppose C /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A2
hence A1 \/ A2 c= A2 by A6, A23, A25, TSEP_1:4; ::_thesis: verum
end;
suppose C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then X0 meets X1 union X2 by A8, TSEP_1:def_3;
then A27: the carrier of (X0 meet (X1 union X2)) = C /\ (A1 \/ A2) by A8, TSEP_1:def_4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, TSEP_1:def_4;
then C /\ (A1 \/ A2) c= A1 /\ A2 by A7, A27, TSEP_1:4;
then A1 \/ A2 c= A2 \/ (A1 /\ A2) by A23, A26, XBOOLE_1:13;
hence A1 \/ A2 c= A2 by XBOOLE_1:12, XBOOLE_1:17; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A2 ; ::_thesis: verum
end;
A28: now__::_thesis:_(_C_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A2_)
assume C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then X0 meets X1 union X2 by A8, TSEP_1:def_3;
then A29: the carrier of (X0 meet (X1 union X2)) = C /\ (A1 \/ A2) by A8, TSEP_1:def_4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, TSEP_1:def_4;
then A30: C /\ (A1 \/ A2) c= A1 /\ A2 by A7, A29, TSEP_1:4;
A31: A1 /\ A2 c= A2 by XBOOLE_1:17;
then A32: C /\ (A1 \/ A2) c= A2 by A30, XBOOLE_1:1;
now__::_thesis:_A1_\/_A2_c=_A2
percases ( C2 /\ (A1 \/ A2) = {} or C2 /\ (A1 \/ A2) <> {} ) ;
suppose C2 /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A2
hence A1 \/ A2 c= A2 by A23, A30, A31, XBOOLE_1:1; ::_thesis: verum
end;
suppose C2 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C2 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y2 meets X1 union X2 by A8, TSEP_1:def_3;
then the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A8, TSEP_1:def_4;
then C2 /\ (A1 \/ A2) c= A2 by A6, TSEP_1:4;
hence A1 \/ A2 c= A2 by A23, A32, XBOOLE_1:8; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A2 ; ::_thesis: verum
end;
A1 c= A1 \/ A2 by XBOOLE_1:7;
then A1 c= A2 by A23, A24, A28, XBOOLE_1:1;
hence contradiction by A2, TSEP_1:4; ::_thesis: verum
end;
hence ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) by A10; ::_thesis: verum
end;
theorem Th33: :: TMAP_1:33
for X being non empty TopSpace
for X1, X2, Y1, Y2, X0 being non empty SubSpace of X st X1 meets X2 & X1 is not SubSpace of X2 & X2 is not SubSpace of X1 & X1 union X2 is not SubSpace of Y1 union Y2 & TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & X0 meet (X1 union X2) is SubSpace of X1 meet X2 holds
( Y1 union Y2 meets X1 union X2 & X0 meets X1 union X2 )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, Y1, Y2, X0 being non empty SubSpace of X st X1 meets X2 & X1 is not SubSpace of X2 & X2 is not SubSpace of X1 & X1 union X2 is not SubSpace of Y1 union Y2 & TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & X0 meet (X1 union X2) is SubSpace of X1 meet X2 holds
( Y1 union Y2 meets X1 union X2 & X0 meets X1 union X2 )
let X1, X2, Y1, Y2, X0 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 & X1 is not SubSpace of X2 & X2 is not SubSpace of X1 & X1 union X2 is not SubSpace of Y1 union Y2 & TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & X0 meet (X1 union X2) is SubSpace of X1 meet X2 implies ( Y1 union Y2 meets X1 union X2 & X0 meets X1 union X2 ) )
assume A1: X1 meets X2 ; ::_thesis: ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or X1 union X2 is SubSpace of Y1 union Y2 or not TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 or not Y1 meet (X1 union X2) is SubSpace of X1 or not Y2 meet (X1 union X2) is SubSpace of X2 or not X0 meet (X1 union X2) is SubSpace of X1 meet X2 or ( Y1 union Y2 meets X1 union X2 & X0 meets X1 union X2 ) )
assume A2: ( X1 is not SubSpace of X2 & X2 is not SubSpace of X1 ) ; ::_thesis: ( X1 union X2 is SubSpace of Y1 union Y2 or not TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 or not Y1 meet (X1 union X2) is SubSpace of X1 or not Y2 meet (X1 union X2) is SubSpace of X2 or not X0 meet (X1 union X2) is SubSpace of X1 meet X2 or ( Y1 union Y2 meets X1 union X2 & X0 meets X1 union X2 ) )
reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider C2 = the carrier of Y2 as Subset of X by TSEP_1:1;
reconsider C1 = the carrier of Y1 as Subset of X by TSEP_1:1;
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
A3: the carrier of (Y1 union Y2) = C1 \/ C2 by TSEP_1:def_2;
A4: Y1 is SubSpace of Y1 union Y2 by TSEP_1:22;
assume A5: X1 union X2 is not SubSpace of Y1 union Y2 ; ::_thesis: ( not TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 or not Y1 meet (X1 union X2) is SubSpace of X1 or not Y2 meet (X1 union X2) is SubSpace of X2 or not X0 meet (X1 union X2) is SubSpace of X1 meet X2 or ( Y1 union Y2 meets X1 union X2 & X0 meets X1 union X2 ) )
assume A6: TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union X0 ; ::_thesis: ( not Y1 meet (X1 union X2) is SubSpace of X1 or not Y2 meet (X1 union X2) is SubSpace of X2 or not X0 meet (X1 union X2) is SubSpace of X1 meet X2 or ( Y1 union Y2 meets X1 union X2 & X0 meets X1 union X2 ) )
assume A7: ( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 ) ; ::_thesis: ( not X0 meet (X1 union X2) is SubSpace of X1 meet X2 or ( Y1 union Y2 meets X1 union X2 & X0 meets X1 union X2 ) )
assume X0 meet (X1 union X2) is SubSpace of X1 meet X2 ; ::_thesis: ( Y1 union Y2 meets X1 union X2 & X0 meets X1 union X2 )
then Y1 meets X1 union X2 by A1, A2, A6, A7, Th32;
hence Y1 union Y2 meets X1 union X2 by A4, Th18; ::_thesis: X0 meets X1 union X2
A8: the carrier of (X1 union X2) = A1 \/ A2 by TSEP_1:def_2;
then A9: not A1 \/ A2 c= C1 \/ C2 by A5, A3, TSEP_1:4;
now__::_thesis:_not_X0_misses_X1_union_X2
assume X0 misses X1 union X2 ; ::_thesis: contradiction
then A10: C misses A1 \/ A2 by A8, TSEP_1:def_3;
the carrier of X = (C1 \/ C2) \/ C by A6, A3, TSEP_1:def_2;
then A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by XBOOLE_1:28
.= ((C1 \/ C2) /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23
.= ((C1 \/ C2) /\ (A1 \/ A2)) \/ {} by A10, XBOOLE_0:def_7
.= (C1 \/ C2) /\ (A1 \/ A2) ;
hence contradiction by A9, XBOOLE_1:17; ::_thesis: verum
end;
hence X0 meets X1 union X2 ; ::_thesis: verum
end;
theorem Th34: :: TMAP_1:34
for X being non empty TopSpace
for X1, X2, X0 being non empty SubSpace of X holds
( ( not X1 union X2 meets X0 or X1 meets X0 or X2 meets X0 ) & ( ( X1 meets X0 or X2 meets X0 ) implies X1 union X2 meets X0 ) & ( not X0 meets X1 union X2 or X0 meets X1 or X0 meets X2 ) & ( ( X0 meets X1 or X0 meets X2 ) implies X0 meets X1 union X2 ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, X0 being non empty SubSpace of X holds
( ( not X1 union X2 meets X0 or X1 meets X0 or X2 meets X0 ) & ( ( X1 meets X0 or X2 meets X0 ) implies X1 union X2 meets X0 ) & ( not X0 meets X1 union X2 or X0 meets X1 or X0 meets X2 ) & ( ( X0 meets X1 or X0 meets X2 ) implies X0 meets X1 union X2 ) )
let X1, X2, X0 be non empty SubSpace of X; ::_thesis: ( ( not X1 union X2 meets X0 or X1 meets X0 or X2 meets X0 ) & ( ( X1 meets X0 or X2 meets X0 ) implies X1 union X2 meets X0 ) & ( not X0 meets X1 union X2 or X0 meets X1 or X0 meets X2 ) & ( ( X0 meets X1 or X0 meets X2 ) implies X0 meets X1 union X2 ) )
reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
A1: ( ( X1 meets X0 or X2 meets X0 ) implies X1 union X2 meets X0 )
proof
assume ( X1 meets X0 or X2 meets X0 ) ; ::_thesis: X1 union X2 meets X0
then ( A1 meets A0 or A2 meets A0 ) by TSEP_1:def_3;
then ( A1 /\ A0 <> {} or A2 /\ A0 <> {} ) by XBOOLE_0:def_7;
then (A1 /\ A0) \/ (A2 /\ A0) <> {} ;
then (A1 \/ A2) /\ A0 <> {} by XBOOLE_1:23;
then the carrier of (X1 union X2) /\ A0 <> {} by TSEP_1:def_2;
then the carrier of (X1 union X2) meets A0 by XBOOLE_0:def_7;
hence X1 union X2 meets X0 by TSEP_1:def_3; ::_thesis: verum
end;
A2: ( not X1 union X2 meets X0 or X1 meets X0 or X2 meets X0 )
proof
assume X1 union X2 meets X0 ; ::_thesis: ( X1 meets X0 or X2 meets X0 )
then the carrier of (X1 union X2) meets A0 by TSEP_1:def_3;
then the carrier of (X1 union X2) /\ A0 <> {} by XBOOLE_0:def_7;
then (A1 \/ A2) /\ A0 <> {} by TSEP_1:def_2;
then (A1 /\ A0) \/ (A2 /\ A0) <> {} by XBOOLE_1:23;
then ( A1 /\ A0 <> {} or A2 /\ A0 <> {} ) ;
then ( A1 meets A0 or A2 meets A0 ) by XBOOLE_0:def_7;
hence ( X1 meets X0 or X2 meets X0 ) by TSEP_1:def_3; ::_thesis: verum
end;
hence ( ( X1 meets X0 or X2 meets X0 ) iff X1 union X2 meets X0 ) by A1; ::_thesis: ( ( X0 meets X1 or X0 meets X2 ) iff X0 meets X1 union X2 )
thus ( ( X0 meets X1 or X0 meets X2 ) iff X0 meets X1 union X2 ) by A2, A1; ::_thesis: verum
end;
theorem :: TMAP_1:35
for X being non empty TopSpace
for X1, X2, X0 being non empty SubSpace of X holds
( ( X1 union X2 misses X0 implies ( X1 misses X0 & X2 misses X0 ) ) & ( X1 misses X0 & X2 misses X0 implies X1 union X2 misses X0 ) & ( X0 misses X1 union X2 implies ( X0 misses X1 & X0 misses X2 ) ) & ( X0 misses X1 & X0 misses X2 implies X0 misses X1 union X2 ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, X0 being non empty SubSpace of X holds
( ( X1 union X2 misses X0 implies ( X1 misses X0 & X2 misses X0 ) ) & ( X1 misses X0 & X2 misses X0 implies X1 union X2 misses X0 ) & ( X0 misses X1 union X2 implies ( X0 misses X1 & X0 misses X2 ) ) & ( X0 misses X1 & X0 misses X2 implies X0 misses X1 union X2 ) )
let X1, X2, X0 be non empty SubSpace of X; ::_thesis: ( ( X1 union X2 misses X0 implies ( X1 misses X0 & X2 misses X0 ) ) & ( X1 misses X0 & X2 misses X0 implies X1 union X2 misses X0 ) & ( X0 misses X1 union X2 implies ( X0 misses X1 & X0 misses X2 ) ) & ( X0 misses X1 & X0 misses X2 implies X0 misses X1 union X2 ) )
reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
A1: ( X1 union X2 misses X0 implies ( X1 misses X0 & X2 misses X0 ) )
proof
assume X1 union X2 misses X0 ; ::_thesis: ( X1 misses X0 & X2 misses X0 )
then the carrier of (X1 union X2) misses A0 by TSEP_1:def_3;
then the carrier of (X1 union X2) /\ A0 = {} by XBOOLE_0:def_7;
then (A1 \/ A2) /\ A0 = {} by TSEP_1:def_2;
then A2: (A1 /\ A0) \/ (A2 /\ A0) = {} by XBOOLE_1:23;
then A2 /\ A0 = {} ;
then A3: A2 misses A0 by XBOOLE_0:def_7;
A1 /\ A0 = {} by A2;
then A1 misses A0 by XBOOLE_0:def_7;
hence ( X1 misses X0 & X2 misses X0 ) by A3, TSEP_1:def_3; ::_thesis: verum
end;
A4: ( X1 misses X0 & X2 misses X0 implies X1 union X2 misses X0 )
proof
assume that
A5: X1 misses X0 and
A6: X2 misses X0 ; ::_thesis: X1 union X2 misses X0
A1 misses A0 by A5, TSEP_1:def_3;
then A7: A1 /\ A0 = {} by XBOOLE_0:def_7;
A2 misses A0 by A6, TSEP_1:def_3;
then (A1 /\ A0) \/ (A2 /\ A0) = {} by A7, XBOOLE_0:def_7;
then (A1 \/ A2) /\ A0 = {} by XBOOLE_1:23;
then the carrier of (X1 union X2) /\ A0 = {} by TSEP_1:def_2;
then the carrier of (X1 union X2) misses A0 by XBOOLE_0:def_7;
hence X1 union X2 misses X0 by TSEP_1:def_3; ::_thesis: verum
end;
hence ( X1 union X2 misses X0 iff ( X1 misses X0 & X2 misses X0 ) ) by A1; ::_thesis: ( X0 misses X1 union X2 iff ( X0 misses X1 & X0 misses X2 ) )
thus ( X0 misses X1 union X2 iff ( X0 misses X1 & X0 misses X2 ) ) by A1, A4; ::_thesis: verum
end;
theorem :: TMAP_1:36
for X being non empty TopSpace
for X1, X2, X0 being non empty SubSpace of X st X1 meets X2 holds
( ( X1 meet X2 meets X0 implies ( X1 meets X0 & X2 meets X0 ) ) & ( X0 meets X1 meet X2 implies ( X0 meets X1 & X0 meets X2 ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, X0 being non empty SubSpace of X st X1 meets X2 holds
( ( X1 meet X2 meets X0 implies ( X1 meets X0 & X2 meets X0 ) ) & ( X0 meets X1 meet X2 implies ( X0 meets X1 & X0 meets X2 ) ) )
let X1, X2, X0 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies ( ( X1 meet X2 meets X0 implies ( X1 meets X0 & X2 meets X0 ) ) & ( X0 meets X1 meet X2 implies ( X0 meets X1 & X0 meets X2 ) ) ) )
reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume A1: X1 meets X2 ; ::_thesis: ( ( X1 meet X2 meets X0 implies ( X1 meets X0 & X2 meets X0 ) ) & ( X0 meets X1 meet X2 implies ( X0 meets X1 & X0 meets X2 ) ) )
thus ( X1 meet X2 meets X0 implies ( X1 meets X0 & X2 meets X0 ) ) ::_thesis: ( X0 meets X1 meet X2 implies ( X0 meets X1 & X0 meets X2 ) )
proof
assume X1 meet X2 meets X0 ; ::_thesis: ( X1 meets X0 & X2 meets X0 )
then the carrier of (X1 meet X2) meets A0 by TSEP_1:def_3;
then the carrier of (X1 meet X2) /\ A0 <> {} by XBOOLE_0:def_7;
then (A1 /\ A2) /\ A0 <> {} by A1, TSEP_1:def_4;
then A2: A1 /\ (A2 /\ (A0 /\ A0)) <> {} by XBOOLE_1:16;
then A1 /\ (A0 /\ (A2 /\ A0)) <> {} by XBOOLE_1:16;
then (A1 /\ A0) /\ (A2 /\ A0) <> {} by XBOOLE_1:16;
then A1 /\ A0 <> {} ;
then A3: A1 meets A0 by XBOOLE_0:def_7;
A2 /\ A0 <> {} by A2;
then A2 meets A0 by XBOOLE_0:def_7;
hence ( X1 meets X0 & X2 meets X0 ) by A3, TSEP_1:def_3; ::_thesis: verum
end;
hence ( X0 meets X1 meet X2 implies ( X0 meets X1 & X0 meets X2 ) ) ; ::_thesis: verum
end;
theorem :: TMAP_1:37
for X being non empty TopSpace
for X1, X2, X0 being non empty SubSpace of X st X1 meets X2 holds
( ( ( X1 misses X0 or X2 misses X0 ) implies X1 meet X2 misses X0 ) & ( ( X0 misses X1 or X0 misses X2 ) implies X0 misses X1 meet X2 ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, X0 being non empty SubSpace of X st X1 meets X2 holds
( ( ( X1 misses X0 or X2 misses X0 ) implies X1 meet X2 misses X0 ) & ( ( X0 misses X1 or X0 misses X2 ) implies X0 misses X1 meet X2 ) )
let X1, X2, X0 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies ( ( ( X1 misses X0 or X2 misses X0 ) implies X1 meet X2 misses X0 ) & ( ( X0 misses X1 or X0 misses X2 ) implies X0 misses X1 meet X2 ) ) )
reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume A1: X1 meets X2 ; ::_thesis: ( ( ( X1 misses X0 or X2 misses X0 ) implies X1 meet X2 misses X0 ) & ( ( X0 misses X1 or X0 misses X2 ) implies X0 misses X1 meet X2 ) )
thus ( ( X1 misses X0 or X2 misses X0 ) implies X1 meet X2 misses X0 ) ::_thesis: ( ( X0 misses X1 or X0 misses X2 ) implies X0 misses X1 meet X2 )
proof
assume ( X1 misses X0 or X2 misses X0 ) ; ::_thesis: X1 meet X2 misses X0
then ( A1 misses A0 or A2 misses A0 ) by TSEP_1:def_3;
then ( A1 /\ A0 = {} or A2 /\ A0 = {} ) by XBOOLE_0:def_7;
then (A1 /\ A0) /\ (A2 /\ A0) = {} ;
then A1 /\ ((A2 /\ A0) /\ A0) = {} by XBOOLE_1:16;
then A1 /\ (A2 /\ (A0 /\ A0)) = {} by XBOOLE_1:16;
then (A1 /\ A2) /\ A0 = {} by XBOOLE_1:16;
then the carrier of (X1 meet X2) /\ A0 = {} by A1, TSEP_1:def_4;
then the carrier of (X1 meet X2) misses A0 by XBOOLE_0:def_7;
hence X1 meet X2 misses X0 by TSEP_1:def_3; ::_thesis: verum
end;
hence ( ( X0 misses X1 or X0 misses X2 ) implies X0 misses X1 meet X2 ) ; ::_thesis: verum
end;
theorem Th38: :: TMAP_1:38
for X being non empty TopSpace
for X1 being non empty SubSpace of X
for X0 being non empty closed SubSpace of X st X0 meets X1 holds
X0 meet X1 is closed SubSpace of X1
proof
let X be non empty TopSpace; ::_thesis: for X1 being non empty SubSpace of X
for X0 being non empty closed SubSpace of X st X0 meets X1 holds
X0 meet X1 is closed SubSpace of X1
let X1 be non empty SubSpace of X; ::_thesis: for X0 being non empty closed SubSpace of X st X0 meets X1 holds
X0 meet X1 is closed SubSpace of X1
let X0 be non empty closed SubSpace of X; ::_thesis: ( X0 meets X1 implies X0 meet X1 is closed SubSpace of X1 )
reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
reconsider B = A0 /\ A1 as Subset of X1 by XBOOLE_1:17;
( B = A0 /\ ([#] X1) & A0 is closed ) by TSEP_1:11;
then A1: B is closed by PRE_TOPC:13;
assume A2: X0 meets X1 ; ::_thesis: X0 meet X1 is closed SubSpace of X1
then B = the carrier of (X0 meet X1) by TSEP_1:def_4;
hence X0 meet X1 is closed SubSpace of X1 by A2, A1, TSEP_1:11, TSEP_1:27; ::_thesis: verum
end;
theorem Th39: :: TMAP_1:39
for X being non empty TopSpace
for X1 being non empty SubSpace of X
for X0 being non empty open SubSpace of X st X0 meets X1 holds
X0 meet X1 is open SubSpace of X1
proof
let X be non empty TopSpace; ::_thesis: for X1 being non empty SubSpace of X
for X0 being non empty open SubSpace of X st X0 meets X1 holds
X0 meet X1 is open SubSpace of X1
let X1 be non empty SubSpace of X; ::_thesis: for X0 being non empty open SubSpace of X st X0 meets X1 holds
X0 meet X1 is open SubSpace of X1
let X0 be non empty open SubSpace of X; ::_thesis: ( X0 meets X1 implies X0 meet X1 is open SubSpace of X1 )
reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
reconsider B = A0 /\ A1 as Subset of X1 by XBOOLE_1:17;
( B = A0 /\ ([#] X1) & A0 is open ) by TSEP_1:16;
then A1: B is open by TOPS_2:24;
assume A2: X0 meets X1 ; ::_thesis: X0 meet X1 is open SubSpace of X1
then B = the carrier of (X0 meet X1) by TSEP_1:def_4;
hence X0 meet X1 is open SubSpace of X1 by A2, A1, TSEP_1:16, TSEP_1:27; ::_thesis: verum
end;
theorem :: TMAP_1:40
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for X0 being non empty closed SubSpace of X st X1 is SubSpace of X0 & X0 misses X2 holds
( X1 is closed SubSpace of X1 union X2 & X1 is closed SubSpace of X2 union X1 )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for X0 being non empty closed SubSpace of X st X1 is SubSpace of X0 & X0 misses X2 holds
( X1 is closed SubSpace of X1 union X2 & X1 is closed SubSpace of X2 union X1 )
let X1, X2 be non empty SubSpace of X; ::_thesis: for X0 being non empty closed SubSpace of X st X1 is SubSpace of X0 & X0 misses X2 holds
( X1 is closed SubSpace of X1 union X2 & X1 is closed SubSpace of X2 union X1 )
A1: X1 is SubSpace of X1 union X2 by TSEP_1:22;
reconsider S = TopStruct(# the carrier of X1, the topology of X1 #) as SubSpace of X by Th6;
let X0 be non empty closed SubSpace of X; ::_thesis: ( X1 is SubSpace of X0 & X0 misses X2 implies ( X1 is closed SubSpace of X1 union X2 & X1 is closed SubSpace of X2 union X1 ) )
assume A2: X1 is SubSpace of X0 ; ::_thesis: ( not X0 misses X2 or ( X1 is closed SubSpace of X1 union X2 & X1 is closed SubSpace of X2 union X1 ) )
assume X0 misses X2 ; ::_thesis: ( X1 is closed SubSpace of X1 union X2 & X1 is closed SubSpace of X2 union X1 )
then A3: X0 meet (X1 union X2) = TopStruct(# the carrier of X1, the topology of X1 #) by A2, Th28;
X0 meets X1 by A2, Th17;
then X0 meets X1 union X2 by A1, Th18;
then S is closed SubSpace of X1 union X2 by A3, Th38;
hence ( X1 is closed SubSpace of X1 union X2 & X1 is closed SubSpace of X2 union X1 ) by Th8; ::_thesis: verum
end;
theorem Th41: :: TMAP_1:41
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for X0 being non empty open SubSpace of X st X1 is SubSpace of X0 & X0 misses X2 holds
( X1 is open SubSpace of X1 union X2 & X1 is open SubSpace of X2 union X1 )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for X0 being non empty open SubSpace of X st X1 is SubSpace of X0 & X0 misses X2 holds
( X1 is open SubSpace of X1 union X2 & X1 is open SubSpace of X2 union X1 )
let X1, X2 be non empty SubSpace of X; ::_thesis: for X0 being non empty open SubSpace of X st X1 is SubSpace of X0 & X0 misses X2 holds
( X1 is open SubSpace of X1 union X2 & X1 is open SubSpace of X2 union X1 )
A1: X1 is SubSpace of X1 union X2 by TSEP_1:22;
reconsider S = TopStruct(# the carrier of X1, the topology of X1 #) as SubSpace of X by Th6;
let X0 be non empty open SubSpace of X; ::_thesis: ( X1 is SubSpace of X0 & X0 misses X2 implies ( X1 is open SubSpace of X1 union X2 & X1 is open SubSpace of X2 union X1 ) )
assume A2: X1 is SubSpace of X0 ; ::_thesis: ( not X0 misses X2 or ( X1 is open SubSpace of X1 union X2 & X1 is open SubSpace of X2 union X1 ) )
assume X0 misses X2 ; ::_thesis: ( X1 is open SubSpace of X1 union X2 & X1 is open SubSpace of X2 union X1 )
then A3: X0 meet (X1 union X2) = TopStruct(# the carrier of X1, the topology of X1 #) by A2, Th28;
X0 meets X1 by A2, Th17;
then X0 meets X1 union X2 by A1, Th18;
then S is open SubSpace of X1 union X2 by A3, Th39;
hence ( X1 is open SubSpace of X1 union X2 & X1 is open SubSpace of X2 union X1 ) by Th9; ::_thesis: verum
end;
begin
definition
let X, Y be non empty TopSpace;
let f be Function of X,Y;
let x be Point of X;
predf is_continuous_at x means :Def2: :: TMAP_1:def 2
for G being a_neighborhood of f . x ex H being a_neighborhood of x st f .: H c= G;
end;
:: deftheorem Def2 defines is_continuous_at TMAP_1:def_2_:_
for X, Y being non empty TopSpace
for f being Function of X,Y
for x being Point of X holds
( f is_continuous_at x iff for G being a_neighborhood of f . x ex H being a_neighborhood of x st f .: H c= G );
notation
let X, Y be non empty TopSpace;
let f be Function of X,Y;
let x be Point of X;
antonym f is_not_continuous_at x for f is_continuous_at x;
end;
theorem Th42: :: TMAP_1:42
for Y, X being non empty TopSpace
for f being Function of X,Y
for x being Point of X holds
( f is_continuous_at x iff for G being a_neighborhood of f . x holds f " G is a_neighborhood of x )
proof
let Y, X be non empty TopSpace; ::_thesis: for f being Function of X,Y
for x being Point of X holds
( f is_continuous_at x iff for G being a_neighborhood of f . x holds f " G is a_neighborhood of x )
let f be Function of X,Y; ::_thesis: for x being Point of X holds
( f is_continuous_at x iff for G being a_neighborhood of f . x holds f " G is a_neighborhood of x )
let x be Point of X; ::_thesis: ( f is_continuous_at x iff for G being a_neighborhood of f . x holds f " G is a_neighborhood of x )
thus ( f is_continuous_at x implies for G being a_neighborhood of f . x holds f " G is a_neighborhood of x ) ::_thesis: ( ( for G being a_neighborhood of f . x holds f " G is a_neighborhood of x ) implies f is_continuous_at x )
proof
assume A1: f is_continuous_at x ; ::_thesis: for G being a_neighborhood of f . x holds f " G is a_neighborhood of x
let G be a_neighborhood of f . x; ::_thesis: f " G is a_neighborhood of x
consider H being a_neighborhood of x such that
A2: f .: H c= G by A1, Def2;
ex V being Subset of X st
( V is open & V c= f " G & x in V )
proof
consider V being Subset of X such that
A3: ( V is open & V c= H & x in V ) by CONNSP_2:6;
take V ; ::_thesis: ( V is open & V c= f " G & x in V )
H c= f " G by A2, FUNCT_2:95;
hence ( V is open & V c= f " G & x in V ) by A3, XBOOLE_1:1; ::_thesis: verum
end;
hence f " G is a_neighborhood of x by CONNSP_2:6; ::_thesis: verum
end;
assume A4: for G being a_neighborhood of f . x holds f " G is a_neighborhood of x ; ::_thesis: f is_continuous_at x
let G be a_neighborhood of f . x; :: according to TMAP_1:def_2 ::_thesis: ex H being a_neighborhood of x st f .: H c= G
reconsider H = f " G as a_neighborhood of x by A4;
take H ; ::_thesis: f .: H c= G
thus f .: H c= G by FUNCT_1:75; ::_thesis: verum
end;
theorem Th43: :: TMAP_1:43
for X, Y being non empty TopSpace
for f being Function of X,Y
for x being Point of X holds
( f is_continuous_at x iff for G being Subset of Y st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y
for x being Point of X holds
( f is_continuous_at x iff for G being Subset of Y st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G ) )
let f be Function of X,Y; ::_thesis: for x being Point of X holds
( f is_continuous_at x iff for G being Subset of Y st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G ) )
let x be Point of X; ::_thesis: ( f is_continuous_at x iff for G being Subset of Y st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G ) )
thus ( f is_continuous_at x implies for G being Subset of Y st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G ) ) ::_thesis: ( ( for G being Subset of Y st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G ) ) implies f is_continuous_at x )
proof
assume A1: f is_continuous_at x ; ::_thesis: for G being Subset of Y st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G )
let G be Subset of Y; ::_thesis: ( G is open & f . x in G implies ex H being Subset of X st
( H is open & x in H & f .: H c= G ) )
assume ( G is open & f . x in G ) ; ::_thesis: ex H being Subset of X st
( H is open & x in H & f .: H c= G )
then reconsider G0 = G as a_neighborhood of f . x by CONNSP_2:3;
consider H0 being a_neighborhood of x such that
A2: f .: H0 c= G0 by A1, Def2;
consider H being Subset of X such that
A3: H is open and
A4: H c= H0 and
A5: x in H by CONNSP_2:6;
take H ; ::_thesis: ( H is open & x in H & f .: H c= G )
f .: H c= f .: H0 by A4, RELAT_1:123;
hence ( H is open & x in H & f .: H c= G ) by A2, A3, A5, XBOOLE_1:1; ::_thesis: verum
end;
assume A6: for G being Subset of Y st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G ) ; ::_thesis: f is_continuous_at x
let G0 be a_neighborhood of f . x; :: according to TMAP_1:def_2 ::_thesis: ex H being a_neighborhood of x st f .: H c= G0
consider G being Subset of Y such that
A7: G is open and
A8: G c= G0 and
A9: f . x in G by CONNSP_2:6;
consider H being Subset of X such that
A10: ( H is open & x in H ) and
A11: f .: H c= G by A6, A7, A9;
reconsider H0 = H as a_neighborhood of x by A10, CONNSP_2:3;
take H0 ; ::_thesis: f .: H0 c= G0
thus f .: H0 c= G0 by A8, A11, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th44: :: TMAP_1:44
for Y, X being non empty TopSpace
for f being Function of X,Y holds
( f is continuous iff for x being Point of X holds f is_continuous_at x )
proof
let Y, X be non empty TopSpace; ::_thesis: for f being Function of X,Y holds
( f is continuous iff for x being Point of X holds f is_continuous_at x )
let f be Function of X,Y; ::_thesis: ( f is continuous iff for x being Point of X holds f is_continuous_at x )
thus ( f is continuous implies for x being Point of X holds f is_continuous_at x ) ::_thesis: ( ( for x being Point of X holds f is_continuous_at x ) implies f is continuous )
proof
assume A1: f is continuous ; ::_thesis: for x being Point of X holds f is_continuous_at x
let x be Point of X; ::_thesis: f is_continuous_at x
for G being a_neighborhood of f . x ex H being a_neighborhood of x st f .: H c= G by A1, BORSUK_1:def_1;
hence f is_continuous_at x by Def2; ::_thesis: verum
end;
assume A2: for x being Point of X holds f is_continuous_at x ; ::_thesis: f is continuous
thus f is continuous ::_thesis: verum
proof
let x be Point of X; :: according to BORSUK_1:def_1 ::_thesis: for b1 being a_neighborhood of f . x ex b2 being a_neighborhood of x st f .: b2 c= b1
let G be a_neighborhood of f . x; ::_thesis: ex b1 being a_neighborhood of x st f .: b1 c= G
f is_continuous_at x by A2;
hence ex b1 being a_neighborhood of x st f .: b1 c= G by Def2; ::_thesis: verum
end;
end;
theorem Th45: :: TMAP_1:45
for X, Y, Z being non empty TopSpace st the carrier of Y = the carrier of Z & the topology of Z c= the topology of Y holds
for f being Function of X,Y
for g being Function of X,Z st f = g holds
for x being Point of X st f is_continuous_at x holds
g is_continuous_at x
proof
let X, Y, Z be non empty TopSpace; ::_thesis: ( the carrier of Y = the carrier of Z & the topology of Z c= the topology of Y implies for f being Function of X,Y
for g being Function of X,Z st f = g holds
for x being Point of X st f is_continuous_at x holds
g is_continuous_at x )
assume that
A1: the carrier of Y = the carrier of Z and
A2: the topology of Z c= the topology of Y ; ::_thesis: for f being Function of X,Y
for g being Function of X,Z st f = g holds
for x being Point of X st f is_continuous_at x holds
g is_continuous_at x
let f be Function of X,Y; ::_thesis: for g being Function of X,Z st f = g holds
for x being Point of X st f is_continuous_at x holds
g is_continuous_at x
let g be Function of X,Z; ::_thesis: ( f = g implies for x being Point of X st f is_continuous_at x holds
g is_continuous_at x )
assume A3: f = g ; ::_thesis: for x being Point of X st f is_continuous_at x holds
g is_continuous_at x
let x be Point of X; ::_thesis: ( f is_continuous_at x implies g is_continuous_at x )
assume A4: f is_continuous_at x ; ::_thesis: g is_continuous_at x
for G being Subset of Z st G is open & g . x in G holds
ex H being Subset of X st
( H is open & x in H & g .: H c= G )
proof
let G be Subset of Z; ::_thesis: ( G is open & g . x in G implies ex H being Subset of X st
( H is open & x in H & g .: H c= G ) )
reconsider F = G as Subset of Y by A1;
assume that
A5: G is open and
A6: g . x in G ; ::_thesis: ex H being Subset of X st
( H is open & x in H & g .: H c= G )
G in the topology of Z by A5, PRE_TOPC:def_2;
then F is open by A2, PRE_TOPC:def_2;
then consider H being Subset of X such that
A7: ( H is open & x in H & f .: H c= F ) by A3, A4, A6, Th43;
take H ; ::_thesis: ( H is open & x in H & g .: H c= G )
thus ( H is open & x in H & g .: H c= G ) by A3, A7; ::_thesis: verum
end;
hence g is_continuous_at x by Th43; ::_thesis: verum
end;
theorem Th46: :: TMAP_1:46
for X, Y, Z being non empty TopSpace st the carrier of X = the carrier of Y & the topology of Y c= the topology of X holds
for f being Function of X,Z
for g being Function of Y,Z st f = g holds
for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x
proof
let X, Y, Z be non empty TopSpace; ::_thesis: ( the carrier of X = the carrier of Y & the topology of Y c= the topology of X implies for f being Function of X,Z
for g being Function of Y,Z st f = g holds
for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x )
assume that
A1: the carrier of X = the carrier of Y and
A2: the topology of Y c= the topology of X ; ::_thesis: for f being Function of X,Z
for g being Function of Y,Z st f = g holds
for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x
let f be Function of X,Z; ::_thesis: for g being Function of Y,Z st f = g holds
for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x
let g be Function of Y,Z; ::_thesis: ( f = g implies for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x )
assume A3: f = g ; ::_thesis: for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x
let x be Point of X; ::_thesis: for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x
let y be Point of Y; ::_thesis: ( x = y & g is_continuous_at y implies f is_continuous_at x )
assume A4: x = y ; ::_thesis: ( not g is_continuous_at y or f is_continuous_at x )
assume A5: g is_continuous_at y ; ::_thesis: f is_continuous_at x
for G being Subset of Z st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G )
proof
let G be Subset of Z; ::_thesis: ( G is open & f . x in G implies ex H being Subset of X st
( H is open & x in H & f .: H c= G ) )
assume ( G is open & f . x in G ) ; ::_thesis: ex H being Subset of X st
( H is open & x in H & f .: H c= G )
then consider H being Subset of Y such that
A6: H is open and
A7: ( y in H & g .: H c= G ) by A3, A4, A5, Th43;
reconsider F = H as Subset of X by A1;
take F ; ::_thesis: ( F is open & x in F & f .: F c= G )
H in the topology of Y by A6, PRE_TOPC:def_2;
hence ( F is open & x in F & f .: F c= G ) by A2, A3, A4, A7, PRE_TOPC:def_2; ::_thesis: verum
end;
hence f is_continuous_at x by Th43; ::_thesis: verum
end;
theorem Th47: :: TMAP_1:47
for Z, X, Y being non empty TopSpace
for f being Function of X,Y
for g being Function of Y,Z
for x being Point of X
for y being Point of Y st y = f . x & f is_continuous_at x & g is_continuous_at y holds
g * f is_continuous_at x
proof
let Z, X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y
for g being Function of Y,Z
for x being Point of X
for y being Point of Y st y = f . x & f is_continuous_at x & g is_continuous_at y holds
g * f is_continuous_at x
let f be Function of X,Y; ::_thesis: for g being Function of Y,Z
for x being Point of X
for y being Point of Y st y = f . x & f is_continuous_at x & g is_continuous_at y holds
g * f is_continuous_at x
let g be Function of Y,Z; ::_thesis: for x being Point of X
for y being Point of Y st y = f . x & f is_continuous_at x & g is_continuous_at y holds
g * f is_continuous_at x
let x be Point of X; ::_thesis: for y being Point of Y st y = f . x & f is_continuous_at x & g is_continuous_at y holds
g * f is_continuous_at x
let y be Point of Y; ::_thesis: ( y = f . x & f is_continuous_at x & g is_continuous_at y implies g * f is_continuous_at x )
assume A1: y = f . x ; ::_thesis: ( not f is_continuous_at x or not g is_continuous_at y or g * f is_continuous_at x )
assume that
A2: f is_continuous_at x and
A3: g is_continuous_at y ; ::_thesis: g * f is_continuous_at x
for G being a_neighborhood of (g * f) . x holds (g * f) " G is a_neighborhood of x
proof
let G be a_neighborhood of (g * f) . x; ::_thesis: (g * f) " G is a_neighborhood of x
(g * f) . x = g . y by A1, FUNCT_2:15;
then g " G is a_neighborhood of f . x by A1, A3, Th42;
then f " (g " G) is a_neighborhood of x by A2, Th42;
hence (g * f) " G is a_neighborhood of x by RELAT_1:146; ::_thesis: verum
end;
hence g * f is_continuous_at x by Th42; ::_thesis: verum
end;
theorem :: TMAP_1:48
for Z, Y, X being non empty TopSpace
for f being Function of X,Y
for g being Function of Y,Z
for y being Point of Y st f is continuous & g is_continuous_at y holds
for x being Point of X st x in f " {y} holds
g * f is_continuous_at x
proof
let Z, Y, X be non empty TopSpace; ::_thesis: for f being Function of X,Y
for g being Function of Y,Z
for y being Point of Y st f is continuous & g is_continuous_at y holds
for x being Point of X st x in f " {y} holds
g * f is_continuous_at x
let f be Function of X,Y; ::_thesis: for g being Function of Y,Z
for y being Point of Y st f is continuous & g is_continuous_at y holds
for x being Point of X st x in f " {y} holds
g * f is_continuous_at x
let g be Function of Y,Z; ::_thesis: for y being Point of Y st f is continuous & g is_continuous_at y holds
for x being Point of X st x in f " {y} holds
g * f is_continuous_at x
let y be Point of Y; ::_thesis: ( f is continuous & g is_continuous_at y implies for x being Point of X st x in f " {y} holds
g * f is_continuous_at x )
assume A1: f is continuous ; ::_thesis: ( not g is_continuous_at y or for x being Point of X st x in f " {y} holds
g * f is_continuous_at x )
assume A2: g is_continuous_at y ; ::_thesis: for x being Point of X st x in f " {y} holds
g * f is_continuous_at x
let x be Point of X; ::_thesis: ( x in f " {y} implies g * f is_continuous_at x )
assume x in f " {y} ; ::_thesis: g * f is_continuous_at x
then {x} is Subset of (f " {y}) by SUBSET_1:41;
then ( dom f = [#] X & Im (f,x) c= {y} ) by FUNCT_2:95, FUNCT_2:def_1;
then A3: {(f . x)} c= {y} by FUNCT_1:59;
f . x in {(f . x)} by TARSKI:def_1;
then A4: f . x = y by A3, TARSKI:def_1;
f is_continuous_at x by A1, Th44;
hence g * f is_continuous_at x by A2, A4, Th47; ::_thesis: verum
end;
theorem :: TMAP_1:49
for Y, Z, X being non empty TopSpace
for f being Function of X,Y
for g being Function of Y,Z
for x being Point of X st f is_continuous_at x & g is continuous holds
g * f is_continuous_at x
proof
let Y, Z, X be non empty TopSpace; ::_thesis: for f being Function of X,Y
for g being Function of Y,Z
for x being Point of X st f is_continuous_at x & g is continuous holds
g * f is_continuous_at x
let f be Function of X,Y; ::_thesis: for g being Function of Y,Z
for x being Point of X st f is_continuous_at x & g is continuous holds
g * f is_continuous_at x
let g be Function of Y,Z; ::_thesis: for x being Point of X st f is_continuous_at x & g is continuous holds
g * f is_continuous_at x
let x be Point of X; ::_thesis: ( f is_continuous_at x & g is continuous implies g * f is_continuous_at x )
assume A1: f is_continuous_at x ; ::_thesis: ( not g is continuous or g * f is_continuous_at x )
assume g is continuous ; ::_thesis: g * f is_continuous_at x
then g is_continuous_at f . x by Th44;
hence g * f is_continuous_at x by A1, Th47; ::_thesis: verum
end;
theorem :: TMAP_1:50
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is continuous Function of X,Y iff for x being Point of X holds f is_continuous_at x ) by Th44;
theorem Th51: :: TMAP_1:51
for X, Y, Z being non empty TopSpace st the carrier of Y = the carrier of Z & the topology of Z c= the topology of Y holds
for f being continuous Function of X,Y holds f is continuous Function of X,Z
proof
let X, Y, Z be non empty TopSpace; ::_thesis: ( the carrier of Y = the carrier of Z & the topology of Z c= the topology of Y implies for f being continuous Function of X,Y holds f is continuous Function of X,Z )
assume that
A1: the carrier of Y = the carrier of Z and
A2: the topology of Z c= the topology of Y ; ::_thesis: for f being continuous Function of X,Y holds f is continuous Function of X,Z
let f be continuous Function of X,Y; ::_thesis: f is continuous Function of X,Z
reconsider g = f as Function of X,Z by A1;
for x being Point of X holds g is_continuous_at x
proof
let x be Point of X; ::_thesis: g is_continuous_at x
f is_continuous_at x by Th44;
hence g is_continuous_at x by A1, A2, Th45; ::_thesis: verum
end;
hence f is continuous Function of X,Z by Th44; ::_thesis: verum
end;
theorem :: TMAP_1:52
for X, Y, Z being non empty TopSpace st the carrier of X = the carrier of Y & the topology of Y c= the topology of X holds
for f being continuous Function of Y,Z holds f is continuous Function of X,Z
proof
let X, Y, Z be non empty TopSpace; ::_thesis: ( the carrier of X = the carrier of Y & the topology of Y c= the topology of X implies for f being continuous Function of Y,Z holds f is continuous Function of X,Z )
assume that
A1: the carrier of X = the carrier of Y and
A2: the topology of Y c= the topology of X ; ::_thesis: for f being continuous Function of Y,Z holds f is continuous Function of X,Z
let f be continuous Function of Y,Z; ::_thesis: f is continuous Function of X,Z
reconsider g = f as Function of X,Z by A1;
for x being Point of X holds g is_continuous_at x
proof
let x be Point of X; ::_thesis: g is_continuous_at x
reconsider y = x as Point of Y by A1;
f is_continuous_at y by Th44;
hence g is_continuous_at x by A1, A2, Th46; ::_thesis: verum
end;
hence f is continuous Function of X,Z by Th44; ::_thesis: verum
end;
Lm1: for A being set holds {} is Function of A,{}
proof
let A be set ; ::_thesis: {} is Function of A,{}
percases ( A = {} or A <> {} ) ;
supposeA1: A = {} ; ::_thesis: {} is Function of A,{}
reconsider f = {} as PartFunc of A,{} by RELSET_1:12;
f is total by A1;
hence {} is Function of A,{} ; ::_thesis: verum
end;
suppose A <> {} ; ::_thesis: {} is Function of A,{}
thus {} is Function of A,{} by FUNCT_2:def_1, RELSET_1:12; ::_thesis: verum
end;
end;
end;
definition
let S, T be 1-sorted ;
let f be Function of S,T;
let R be 1-sorted ;
assume A1: the carrier of R c= the carrier of S ;
funcf | R -> Function of R,T equals :Def3: :: TMAP_1:def 3
f | the carrier of R;
coherence
f | the carrier of R is Function of R,T
proof
percases not ( the carrier of T = {} & not the carrier of S = {} & not ( the carrier of T = {} & the carrier of S <> {} ) ) ;
suppose ( the carrier of T = {} implies the carrier of S = {} ) ; ::_thesis: f | the carrier of R is Function of R,T
hence f | the carrier of R is Function of R,T by A1, FUNCT_2:32; ::_thesis: verum
end;
supposeA1: ( the carrier of T = {} & the carrier of S <> {} ) ; ::_thesis: f | the carrier of R is Function of R,T
then f | the carrier of R = {} ;
hence f | the carrier of R is Function of R,T by A1, Lm1; ::_thesis: verum
end;
end;
end;
end;
:: deftheorem Def3 defines | TMAP_1:def_3_:_
for S, T being 1-sorted
for f being Function of S,T
for R being 1-sorted st the carrier of R c= the carrier of S holds
f | R = f | the carrier of R;
definition
let X, Y be non empty TopSpace;
let f be Function of X,Y;
let X0 be SubSpace of X;
redefine func f | X0 equals :: TMAP_1:def 4
f | the carrier of X0;
compatibility
for b1 being Function of X0,Y holds
( b1 = f | X0 iff b1 = f | the carrier of X0 )
proof
[#] X0 c= [#] X by PRE_TOPC:def_4;
hence for b1 being Function of X0,Y holds
( b1 = f | X0 iff b1 = f | the carrier of X0 ) by Def3; ::_thesis: verum
end;
end;
:: deftheorem defines | TMAP_1:def_4_:_
for X, Y being non empty TopSpace
for f being Function of X,Y
for X0 being SubSpace of X holds f | X0 = f | the carrier of X0;
theorem Th53: :: TMAP_1:53
for Y, X being non empty TopSpace
for X0 being non empty SubSpace of X
for f being Function of X,Y
for f0 being Function of X0,Y st ( for x being Point of X st x in the carrier of X0 holds
f . x = f0 . x ) holds
f | X0 = f0
proof
let Y, X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for f being Function of X,Y
for f0 being Function of X0,Y st ( for x being Point of X st x in the carrier of X0 holds
f . x = f0 . x ) holds
f | X0 = f0
let X0 be non empty SubSpace of X; ::_thesis: for f being Function of X,Y
for f0 being Function of X0,Y st ( for x being Point of X st x in the carrier of X0 holds
f . x = f0 . x ) holds
f | X0 = f0
let f be Function of X,Y; ::_thesis: for f0 being Function of X0,Y st ( for x being Point of X st x in the carrier of X0 holds
f . x = f0 . x ) holds
f | X0 = f0
let f0 be Function of X0,Y; ::_thesis: ( ( for x being Point of X st x in the carrier of X0 holds
f . x = f0 . x ) implies f | X0 = f0 )
the carrier of X0 is Subset of X by TSEP_1:1;
hence ( ( for x being Point of X st x in the carrier of X0 holds
f . x = f0 . x ) implies f | X0 = f0 ) by FUNCT_2:96; ::_thesis: verum
end;
theorem Th54: :: TMAP_1:54
for Y, X being non empty TopSpace
for X0 being non empty SubSpace of X
for f being Function of X,Y st TopStruct(# the carrier of X0, the topology of X0 #) = TopStruct(# the carrier of X, the topology of X #) holds
f = f | X0
proof
let Y, X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for f being Function of X,Y st TopStruct(# the carrier of X0, the topology of X0 #) = TopStruct(# the carrier of X, the topology of X #) holds
f = f | X0
let X0 be non empty SubSpace of X; ::_thesis: for f being Function of X,Y st TopStruct(# the carrier of X0, the topology of X0 #) = TopStruct(# the carrier of X, the topology of X #) holds
f = f | X0
let f be Function of X,Y; ::_thesis: ( TopStruct(# the carrier of X0, the topology of X0 #) = TopStruct(# the carrier of X, the topology of X #) implies f = f | X0 )
assume TopStruct(# the carrier of X0, the topology of X0 #) = TopStruct(# the carrier of X, the topology of X #) ; ::_thesis: f = f | X0
hence f | X0 = f * (id the carrier of X) by RELAT_1:65
.= f by FUNCT_2:17 ;
::_thesis: verum
end;
theorem :: TMAP_1:55
for Y, X being non empty TopSpace
for X0 being non empty SubSpace of X
for f being Function of X,Y
for A being Subset of X st A c= the carrier of X0 holds
f .: A = (f | X0) .: A by FUNCT_2:97;
theorem :: TMAP_1:56
for X, Y being non empty TopSpace
for X0 being non empty SubSpace of X
for f being Function of X,Y
for B being Subset of Y st f " B c= the carrier of X0 holds
f " B = (f | X0) " B by FUNCT_2:98;
theorem Th57: :: TMAP_1:57
for Y, X being non empty TopSpace
for X0 being non empty SubSpace of X
for g being Function of X0,Y ex h being Function of X,Y st h | X0 = g
proof
let Y, X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for g being Function of X0,Y ex h being Function of X,Y st h | X0 = g
let X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y ex h being Function of X,Y st h | X0 = g
let g be Function of X0,Y; ::_thesis: ex h being Function of X,Y st h | X0 = g
now__::_thesis:_ex_h_being_Function_of_X,Y_st_h_|_X0_=_g
percases ( TopStruct(# the carrier of X, the topology of X #) = TopStruct(# the carrier of X0, the topology of X0 #) or TopStruct(# the carrier of X, the topology of X #) <> TopStruct(# the carrier of X0, the topology of X0 #) ) ;
supposeA1: TopStruct(# the carrier of X, the topology of X #) = TopStruct(# the carrier of X0, the topology of X0 #) ; ::_thesis: ex h being Function of X,Y st h | X0 = g
then reconsider h = g as Function of X,Y ;
take h = h; ::_thesis: h | X0 = g
thus h | X0 = g by A1, Th54; ::_thesis: verum
end;
supposeA2: TopStruct(# the carrier of X, the topology of X #) <> TopStruct(# the carrier of X0, the topology of X0 #) ; ::_thesis: ex h being Function of X,Y st h | X0 = g
Y is SubSpace of Y by TSEP_1:2;
then reconsider B = the carrier of Y as non empty Subset of Y by TSEP_1:1;
set y = the Element of B;
reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
A3: X is SubSpace of X by TSEP_1:2;
then reconsider A = the carrier of X as non empty Subset of X by TSEP_1:1;
reconsider A1 = A \ A0 as Subset of X ;
A4: A0 misses A1 by XBOOLE_1:79;
A0 <> A by A2, A3, TSEP_1:5;
then not A c= A0 by XBOOLE_0:def_10;
then reconsider A1 = A1 as non empty Subset of A by XBOOLE_1:37;
reconsider g1 = A1 --> the Element of B as Function of A1,B ;
reconsider A0 = A0 as non empty Subset of A ;
reconsider g0 = g as Function of A0,B ;
set G = g0 union g1;
the carrier of X = A1 \/ A0 by XBOOLE_1:45;
then reconsider h = g0 union g1 as Function of X,Y ;
take h = h; ::_thesis: h | X0 = g
thus h | X0 = g by A4, Th1; ::_thesis: verum
end;
end;
end;
hence ex h being Function of X,Y st h | X0 = g ; ::_thesis: verum
end;
theorem Th58: :: TMAP_1:58
for Y, X being non empty TopSpace
for f being Function of X,Y
for X0 being non empty SubSpace of X
for x being Point of X
for x0 being Point of X0 st x = x0 & f is_continuous_at x holds
f | X0 is_continuous_at x0
proof
let Y, X be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X0 being non empty SubSpace of X
for x being Point of X
for x0 being Point of X0 st x = x0 & f is_continuous_at x holds
f | X0 is_continuous_at x0
let f be Function of X,Y; ::_thesis: for X0 being non empty SubSpace of X
for x being Point of X
for x0 being Point of X0 st x = x0 & f is_continuous_at x holds
f | X0 is_continuous_at x0
let X0 be non empty SubSpace of X; ::_thesis: for x being Point of X
for x0 being Point of X0 st x = x0 & f is_continuous_at x holds
f | X0 is_continuous_at x0
let x be Point of X; ::_thesis: for x0 being Point of X0 st x = x0 & f is_continuous_at x holds
f | X0 is_continuous_at x0
let x0 be Point of X0; ::_thesis: ( x = x0 & f is_continuous_at x implies f | X0 is_continuous_at x0 )
assume A1: x = x0 ; ::_thesis: ( not f is_continuous_at x or f | X0 is_continuous_at x0 )
assume A2: f is_continuous_at x ; ::_thesis: f | X0 is_continuous_at x0
for G being Subset of Y st G is open & (f | X0) . x0 in G holds
ex H0 being Subset of X0 st
( H0 is open & x0 in H0 & (f | X0) .: H0 c= G )
proof
reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
let G be Subset of Y; ::_thesis: ( G is open & (f | X0) . x0 in G implies ex H0 being Subset of X0 st
( H0 is open & x0 in H0 & (f | X0) .: H0 c= G ) )
assume that
A3: G is open and
A4: (f | X0) . x0 in G ; ::_thesis: ex H0 being Subset of X0 st
( H0 is open & x0 in H0 & (f | X0) .: H0 c= G )
f . x in G by A1, A4, FUNCT_1:49;
then consider H being Subset of X such that
A5: ( H is open & x in H ) and
A6: f .: H c= G by A2, A3, Th43;
reconsider H0 = H /\ C as Subset of X0 by XBOOLE_1:17;
( f .: H0 c= (f .: H) /\ (f .: C) & (f .: H) /\ (f .: C) c= f .: H ) by RELAT_1:121, XBOOLE_1:17;
then f .: H0 c= f .: H by XBOOLE_1:1;
then A7: f .: H0 c= G by A6, XBOOLE_1:1;
take H0 ; ::_thesis: ( H0 is open & x0 in H0 & (f | X0) .: H0 c= G )
( H0 = H /\ ([#] X0) & (f | X0) .: H0 c= f .: H0 ) by RELAT_1:128;
hence ( H0 is open & x0 in H0 & (f | X0) .: H0 c= G ) by A1, A5, A7, TOPS_2:24, XBOOLE_0:def_4, XBOOLE_1:1; ::_thesis: verum
end;
hence f | X0 is_continuous_at x0 by Th43; ::_thesis: verum
end;
theorem Th59: :: TMAP_1:59
for Y, X being non empty TopSpace
for f being Function of X,Y
for X0 being non empty SubSpace of X
for A being Subset of X
for x being Point of X
for x0 being Point of X0 st A c= the carrier of X0 & A is a_neighborhood of x & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
proof
let Y, X be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X0 being non empty SubSpace of X
for A being Subset of X
for x being Point of X
for x0 being Point of X0 st A c= the carrier of X0 & A is a_neighborhood of x & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let f be Function of X,Y; ::_thesis: for X0 being non empty SubSpace of X
for A being Subset of X
for x being Point of X
for x0 being Point of X0 st A c= the carrier of X0 & A is a_neighborhood of x & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let X0 be non empty SubSpace of X; ::_thesis: for A being Subset of X
for x being Point of X
for x0 being Point of X0 st A c= the carrier of X0 & A is a_neighborhood of x & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let A be Subset of X; ::_thesis: for x being Point of X
for x0 being Point of X0 st A c= the carrier of X0 & A is a_neighborhood of x & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let x be Point of X; ::_thesis: for x0 being Point of X0 st A c= the carrier of X0 & A is a_neighborhood of x & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let x0 be Point of X0; ::_thesis: ( A c= the carrier of X0 & A is a_neighborhood of x & x = x0 implies ( f is_continuous_at x iff f | X0 is_continuous_at x0 ) )
assume that
A1: A c= the carrier of X0 and
A2: A is a_neighborhood of x and
A3: x = x0 ; ::_thesis: ( f is_continuous_at x iff f | X0 is_continuous_at x0 )
thus ( f is_continuous_at x implies f | X0 is_continuous_at x0 ) by A3, Th58; ::_thesis: ( f | X0 is_continuous_at x0 implies f is_continuous_at x )
thus ( f | X0 is_continuous_at x0 implies f is_continuous_at x ) ::_thesis: verum
proof
assume A4: f | X0 is_continuous_at x0 ; ::_thesis: f is_continuous_at x
for G being Subset of Y st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G )
proof
let G be Subset of Y; ::_thesis: ( G is open & f . x in G implies ex H being Subset of X st
( H is open & x in H & f .: H c= G ) )
assume that
A5: G is open and
A6: f . x in G ; ::_thesis: ex H being Subset of X st
( H is open & x in H & f .: H c= G )
(f | X0) . x0 in G by A3, A6, FUNCT_1:49;
then consider H0 being Subset of X0 such that
A7: H0 is open and
A8: x0 in H0 and
A9: (f | X0) .: H0 c= G by A4, A5, Th43;
consider V being Subset of X such that
A10: V is open and
A11: V c= A and
A12: x in V by A2, CONNSP_2:6;
reconsider V0 = V as Subset of X0 by A1, A11, XBOOLE_1:1;
A13: H0 /\ V0 c= V by XBOOLE_1:17;
then reconsider H = H0 /\ V0 as Subset of X by XBOOLE_1:1;
A14: for z being Point of Y st z in f .: H holds
z in G
proof
set g = f | X0;
let z be Point of Y; ::_thesis: ( z in f .: H implies z in G )
assume z in f .: H ; ::_thesis: z in G
then consider y being Point of X such that
A15: y in H and
A16: z = f . y by FUNCT_2:65;
y in V by A13, A15;
then y in A by A11;
then A17: z = (f | X0) . y by A1, A16, FUNCT_1:49;
H0 /\ V0 c= H0 by XBOOLE_1:17;
then z in (f | X0) .: H0 by A15, A17, FUNCT_2:35;
hence z in G by A9; ::_thesis: verum
end;
take H ; ::_thesis: ( H is open & x in H & f .: H c= G )
V0 is open by A10, TOPS_2:25;
then H0 /\ V0 is open by A7;
hence ( H is open & x in H & f .: H c= G ) by A3, A8, A10, A12, A13, A14, SUBSET_1:2, TSEP_1:9, XBOOLE_0:def_4; ::_thesis: verum
end;
hence f is_continuous_at x by Th43; ::_thesis: verum
end;
end;
theorem Th60: :: TMAP_1:60
for Y, X being non empty TopSpace
for f being Function of X,Y
for X0 being non empty SubSpace of X
for A being Subset of X
for x being Point of X
for x0 being Point of X0 st A is open & x in A & A c= the carrier of X0 & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
proof
let Y, X be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X0 being non empty SubSpace of X
for A being Subset of X
for x being Point of X
for x0 being Point of X0 st A is open & x in A & A c= the carrier of X0 & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let f be Function of X,Y; ::_thesis: for X0 being non empty SubSpace of X
for A being Subset of X
for x being Point of X
for x0 being Point of X0 st A is open & x in A & A c= the carrier of X0 & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let X0 be non empty SubSpace of X; ::_thesis: for A being Subset of X
for x being Point of X
for x0 being Point of X0 st A is open & x in A & A c= the carrier of X0 & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let A be Subset of X; ::_thesis: for x being Point of X
for x0 being Point of X0 st A is open & x in A & A c= the carrier of X0 & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let x be Point of X; ::_thesis: for x0 being Point of X0 st A is open & x in A & A c= the carrier of X0 & x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let x0 be Point of X0; ::_thesis: ( A is open & x in A & A c= the carrier of X0 & x = x0 implies ( f is_continuous_at x iff f | X0 is_continuous_at x0 ) )
assume that
A1: ( A is open & x in A ) and
A2: A c= the carrier of X0 and
A3: x = x0 ; ::_thesis: ( f is_continuous_at x iff f | X0 is_continuous_at x0 )
thus ( f is_continuous_at x implies f | X0 is_continuous_at x0 ) by A3, Th58; ::_thesis: ( f | X0 is_continuous_at x0 implies f is_continuous_at x )
thus ( f | X0 is_continuous_at x0 implies f is_continuous_at x ) ::_thesis: verum
proof
assume A4: f | X0 is_continuous_at x0 ; ::_thesis: f is_continuous_at x
A is a_neighborhood of x by A1, CONNSP_2:3;
hence f is_continuous_at x by A2, A3, A4, Th59; ::_thesis: verum
end;
end;
theorem :: TMAP_1:61
for Y, X being non empty TopSpace
for f being Function of X,Y
for X0 being non empty open SubSpace of X
for x being Point of X
for x0 being Point of X0 st x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
proof
let Y, X be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X0 being non empty open SubSpace of X
for x being Point of X
for x0 being Point of X0 st x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let f be Function of X,Y; ::_thesis: for X0 being non empty open SubSpace of X
for x being Point of X
for x0 being Point of X0 st x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let X0 be non empty open SubSpace of X; ::_thesis: for x being Point of X
for x0 being Point of X0 st x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let x be Point of X; ::_thesis: for x0 being Point of X0 st x = x0 holds
( f is_continuous_at x iff f | X0 is_continuous_at x0 )
let x0 be Point of X0; ::_thesis: ( x = x0 implies ( f is_continuous_at x iff f | X0 is_continuous_at x0 ) )
assume A1: x = x0 ; ::_thesis: ( f is_continuous_at x iff f | X0 is_continuous_at x0 )
hence ( f is_continuous_at x implies f | X0 is_continuous_at x0 ) by Th58; ::_thesis: ( f | X0 is_continuous_at x0 implies f is_continuous_at x )
thus ( f | X0 is_continuous_at x0 implies f is_continuous_at x ) ::_thesis: verum
proof
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
assume A2: f | X0 is_continuous_at x0 ; ::_thesis: f is_continuous_at x
A is open by TSEP_1:16;
hence f is_continuous_at x by A1, A2, Th60; ::_thesis: verum
end;
end;
registration
let X, Y be non empty TopSpace;
let f be continuous Function of X,Y;
let X0 be non empty SubSpace of X;
clusterf | X0 -> continuous ;
coherence
f | X0 is continuous
proof
for x0 being Point of X0 holds f | X0 is_continuous_at x0
proof
let x0 be Point of X0; ::_thesis: f | X0 is_continuous_at x0
( the carrier of X0 c= the carrier of X & x0 in the carrier of X0 ) by BORSUK_1:1;
then reconsider x = x0 as Point of X ;
f is_continuous_at x by Th44;
hence f | X0 is_continuous_at x0 by Th58; ::_thesis: verum
end;
hence f | X0 is continuous by Th44; ::_thesis: verum
end;
end;
theorem Th62: :: TMAP_1:62
for X, Y, Z being non empty TopSpace
for X0 being non empty SubSpace of X
for f being Function of X,Y
for g being Function of Y,Z holds (g * f) | X0 = g * (f | X0)
proof
let X, Y, Z be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for f being Function of X,Y
for g being Function of Y,Z holds (g * f) | X0 = g * (f | X0)
let X0 be non empty SubSpace of X; ::_thesis: for f being Function of X,Y
for g being Function of Y,Z holds (g * f) | X0 = g * (f | X0)
let f be Function of X,Y; ::_thesis: for g being Function of Y,Z holds (g * f) | X0 = g * (f | X0)
let g be Function of Y,Z; ::_thesis: (g * f) | X0 = g * (f | X0)
set h = g * f;
(g * f) | X0 = (g * f) | the carrier of X0 ;
then reconsider G = (g * f) | the carrier of X0 as Function of X0,Z ;
f | X0 = f | the carrier of X0 ;
then reconsider F0 = f | the carrier of X0 as Function of X0,Y ;
set F = g * F0;
for x being Point of X0 holds G . x = (g * F0) . x
proof
let x be Point of X0; ::_thesis: G . x = (g * F0) . x
the carrier of X0 c= the carrier of X by BORSUK_1:1;
then reconsider y = x as Element of X by TARSKI:def_3;
thus G . x = (g * f) . y by FUNCT_1:49
.= g . (f . y) by FUNCT_2:15
.= g . (F0 . x) by FUNCT_1:49
.= (g * F0) . x by FUNCT_2:15 ; ::_thesis: verum
end;
hence (g * f) | X0 = g * (f | X0) by FUNCT_2:63; ::_thesis: verum
end;
theorem Th63: :: TMAP_1:63
for X, Y, Z being non empty TopSpace
for X0 being non empty SubSpace of X
for g being Function of Y,Z
for f being Function of X,Y st g is continuous & f | X0 is continuous holds
(g * f) | X0 is continuous
proof
let X, Y, Z be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for g being Function of Y,Z
for f being Function of X,Y st g is continuous & f | X0 is continuous holds
(g * f) | X0 is continuous
let X0 be non empty SubSpace of X; ::_thesis: for g being Function of Y,Z
for f being Function of X,Y st g is continuous & f | X0 is continuous holds
(g * f) | X0 is continuous
let g be Function of Y,Z; ::_thesis: for f being Function of X,Y st g is continuous & f | X0 is continuous holds
(g * f) | X0 is continuous
let f be Function of X,Y; ::_thesis: ( g is continuous & f | X0 is continuous implies (g * f) | X0 is continuous )
assume A1: ( g is continuous & f | X0 is continuous ) ; ::_thesis: (g * f) | X0 is continuous
(g * f) | X0 = g * (f | X0) by Th62;
hence (g * f) | X0 is continuous by A1; ::_thesis: verum
end;
theorem :: TMAP_1:64
for X, Y, Z being non empty TopSpace
for X0 being non empty SubSpace of X
for g being continuous Function of Y,Z
for f being Function of X,Y st f | X0 is continuous Function of X0,Y holds
(g * f) | X0 is continuous Function of X0,Z by Th63;
definition
let X, Y be non empty TopSpace;
let X0, X1 be SubSpace of X;
let g be Function of X0,Y;
assume A1: X1 is SubSpace of X0 ;
funcg | X1 -> Function of X1,Y equals :Def5: :: TMAP_1:def 5
g | the carrier of X1;
coherence
g | the carrier of X1 is Function of X1,Y
proof
the carrier of X1 c= the carrier of X0 by A1, TSEP_1:4;
hence g | the carrier of X1 is Function of X1,Y by FUNCT_2:32; ::_thesis: verum
end;
end;
:: deftheorem Def5 defines | TMAP_1:def_5_:_
for X, Y being non empty TopSpace
for X0, X1 being SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
g | X1 = g | the carrier of X1;
theorem Th65: :: TMAP_1:65
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = (g | X1) . x0
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = (g | X1) . x0
let X1, X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y st X1 is SubSpace of X0 holds
for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = (g | X1) . x0
let g be Function of X0,Y; ::_thesis: ( X1 is SubSpace of X0 implies for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = (g | X1) . x0 )
assume A1: X1 is SubSpace of X0 ; ::_thesis: for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = (g | X1) . x0
let x0 be Point of X0; ::_thesis: ( x0 in the carrier of X1 implies g . x0 = (g | X1) . x0 )
assume x0 in the carrier of X1 ; ::_thesis: g . x0 = (g | X1) . x0
hence g . x0 = (g | the carrier of X1) . x0 by FUNCT_1:49
.= (g | X1) . x0 by A1, Def5 ;
::_thesis: verum
end;
theorem :: TMAP_1:66
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for g1 being Function of X1,Y st ( for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = g1 . x0 ) holds
g | X1 = g1
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for g1 being Function of X1,Y st ( for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = g1 . x0 ) holds
g | X1 = g1
let X1, X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y st X1 is SubSpace of X0 holds
for g1 being Function of X1,Y st ( for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = g1 . x0 ) holds
g | X1 = g1
let g be Function of X0,Y; ::_thesis: ( X1 is SubSpace of X0 implies for g1 being Function of X1,Y st ( for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = g1 . x0 ) holds
g | X1 = g1 )
assume A1: X1 is SubSpace of X0 ; ::_thesis: for g1 being Function of X1,Y st ( for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = g1 . x0 ) holds
g | X1 = g1
then A2: the carrier of X1 is Subset of X0 by TSEP_1:1;
let g1 be Function of X1,Y; ::_thesis: ( ( for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = g1 . x0 ) implies g | X1 = g1 )
assume for x0 being Point of X0 st x0 in the carrier of X1 holds
g . x0 = g1 . x0 ; ::_thesis: g | X1 = g1
then g | the carrier of X1 = g1 by A2, FUNCT_2:96;
hence g | X1 = g1 by A1, Def5; ::_thesis: verum
end;
theorem Th67: :: TMAP_1:67
for X, Y being non empty TopSpace
for X0 being non empty SubSpace of X
for g being Function of X0,Y holds g = g | X0
proof
let X, Y be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for g being Function of X0,Y holds g = g | X0
let X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y holds g = g | X0
let g be Function of X0,Y; ::_thesis: g = g | X0
X0 is SubSpace of X0 by TSEP_1:2;
hence g | X0 = g | the carrier of X0 by Def5
.= g * (id the carrier of X0) by RELAT_1:65
.= g by FUNCT_2:17 ;
::_thesis: verum
end;
theorem Th68: :: TMAP_1:68
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X0 st A c= the carrier of X1 holds
g .: A = (g | X1) .: A
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X0 st A c= the carrier of X1 holds
g .: A = (g | X1) .: A
let X1, X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X0 st A c= the carrier of X1 holds
g .: A = (g | X1) .: A
let g be Function of X0,Y; ::_thesis: ( X1 is SubSpace of X0 implies for A being Subset of X0 st A c= the carrier of X1 holds
g .: A = (g | X1) .: A )
assume A1: X1 is SubSpace of X0 ; ::_thesis: for A being Subset of X0 st A c= the carrier of X1 holds
g .: A = (g | X1) .: A
let A be Subset of X0; ::_thesis: ( A c= the carrier of X1 implies g .: A = (g | X1) .: A )
assume A c= the carrier of X1 ; ::_thesis: g .: A = (g | X1) .: A
hence g .: A = (g | the carrier of X1) .: A by FUNCT_2:97
.= (g | X1) .: A by A1, Def5 ;
::_thesis: verum
end;
theorem :: TMAP_1:69
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for B being Subset of Y st g " B c= the carrier of X1 holds
g " B = (g | X1) " B
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for B being Subset of Y st g " B c= the carrier of X1 holds
g " B = (g | X1) " B
let X1, X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y st X1 is SubSpace of X0 holds
for B being Subset of Y st g " B c= the carrier of X1 holds
g " B = (g | X1) " B
let g be Function of X0,Y; ::_thesis: ( X1 is SubSpace of X0 implies for B being Subset of Y st g " B c= the carrier of X1 holds
g " B = (g | X1) " B )
assume A1: X1 is SubSpace of X0 ; ::_thesis: for B being Subset of Y st g " B c= the carrier of X1 holds
g " B = (g | X1) " B
let B be Subset of Y; ::_thesis: ( g " B c= the carrier of X1 implies g " B = (g | X1) " B )
assume g " B c= the carrier of X1 ; ::_thesis: g " B = (g | X1) " B
hence g " B = (g | the carrier of X1) " B by FUNCT_2:98
.= (g | X1) " B by A1, Def5 ;
::_thesis: verum
end;
theorem Th70: :: TMAP_1:70
for X, Y being non empty TopSpace
for X0, X1 being non empty SubSpace of X
for f being Function of X,Y
for g being Function of X0,Y st g = f | X0 & X1 is SubSpace of X0 holds
g | X1 = f | X1
proof
let X, Y be non empty TopSpace; ::_thesis: for X0, X1 being non empty SubSpace of X
for f being Function of X,Y
for g being Function of X0,Y st g = f | X0 & X1 is SubSpace of X0 holds
g | X1 = f | X1
let X0, X1 be non empty SubSpace of X; ::_thesis: for f being Function of X,Y
for g being Function of X0,Y st g = f | X0 & X1 is SubSpace of X0 holds
g | X1 = f | X1
let f be Function of X,Y; ::_thesis: for g being Function of X0,Y st g = f | X0 & X1 is SubSpace of X0 holds
g | X1 = f | X1
let g be Function of X0,Y; ::_thesis: ( g = f | X0 & X1 is SubSpace of X0 implies g | X1 = f | X1 )
assume A1: g = f | X0 ; ::_thesis: ( not X1 is SubSpace of X0 or g | X1 = f | X1 )
assume A2: X1 is SubSpace of X0 ; ::_thesis: g | X1 = f | X1
then A3: the carrier of X1 c= the carrier of X0 by TSEP_1:4;
thus g | X1 = g | the carrier of X1 by A2, Def5
.= f | X1 by A1, A3, FUNCT_1:51 ; ::_thesis: verum
end;
theorem Th71: :: TMAP_1:71
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for f being Function of X,Y st X1 is SubSpace of X0 holds
(f | X0) | X1 = f | X1
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for f being Function of X,Y st X1 is SubSpace of X0 holds
(f | X0) | X1 = f | X1
let X1, X0 be non empty SubSpace of X; ::_thesis: for f being Function of X,Y st X1 is SubSpace of X0 holds
(f | X0) | X1 = f | X1
let f be Function of X,Y; ::_thesis: ( X1 is SubSpace of X0 implies (f | X0) | X1 = f | X1 )
assume A1: X1 is SubSpace of X0 ; ::_thesis: (f | X0) | X1 = f | X1
then A2: the carrier of X1 c= the carrier of X0 by TSEP_1:4;
f | X0 = f | the carrier of X0 ;
then reconsider h = f | the carrier of X0 as Function of X0,Y ;
thus (f | X0) | X1 = h | the carrier of X1 by A1, Def5
.= f | X1 by A2, FUNCT_1:51 ; ::_thesis: verum
end;
theorem Th72: :: TMAP_1:72
for X, Y being non empty TopSpace
for X0, X1, X2 being non empty SubSpace of X st X1 is SubSpace of X0 & X2 is SubSpace of X1 holds
for g being Function of X0,Y holds (g | X1) | X2 = g | X2
proof
let X, Y be non empty TopSpace; ::_thesis: for X0, X1, X2 being non empty SubSpace of X st X1 is SubSpace of X0 & X2 is SubSpace of X1 holds
for g being Function of X0,Y holds (g | X1) | X2 = g | X2
let X0, X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 is SubSpace of X0 & X2 is SubSpace of X1 implies for g being Function of X0,Y holds (g | X1) | X2 = g | X2 )
assume that
A1: X1 is SubSpace of X0 and
A2: X2 is SubSpace of X1 ; ::_thesis: for g being Function of X0,Y holds (g | X1) | X2 = g | X2
A3: X2 is SubSpace of X0 by A1, A2, TSEP_1:7;
let g be Function of X0,Y; ::_thesis: (g | X1) | X2 = g | X2
set h = g | X1;
A4: ( g | X1 = g | the carrier of X1 & the carrier of X2 c= the carrier of X1 ) by A1, A2, Def5, TSEP_1:4;
thus (g | X1) | X2 = (g | X1) | the carrier of X2 by A2, Def5
.= g | the carrier of X2 by A4, FUNCT_1:51
.= g | X2 by A3, Def5 ; ::_thesis: verum
end;
theorem :: TMAP_1:73
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for f being Function of X,Y
for f0 being Function of X1,Y
for g being Function of X0,Y st X0 = X & f = g holds
( g | X1 = f0 iff f | X1 = f0 ) by Def5;
theorem Th74: :: TMAP_1:74
for X, Y being non empty TopSpace
for X0, X1 being non empty SubSpace of X
for g being Function of X0,Y
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & X1 is SubSpace of X0 & g is_continuous_at x0 holds
g | X1 is_continuous_at x1
proof
let X, Y be non empty TopSpace; ::_thesis: for X0, X1 being non empty SubSpace of X
for g being Function of X0,Y
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & X1 is SubSpace of X0 & g is_continuous_at x0 holds
g | X1 is_continuous_at x1
let X0, X1 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & X1 is SubSpace of X0 & g is_continuous_at x0 holds
g | X1 is_continuous_at x1
let g be Function of X0,Y; ::_thesis: for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & X1 is SubSpace of X0 & g is_continuous_at x0 holds
g | X1 is_continuous_at x1
let x0 be Point of X0; ::_thesis: for x1 being Point of X1 st x0 = x1 & X1 is SubSpace of X0 & g is_continuous_at x0 holds
g | X1 is_continuous_at x1
let x1 be Point of X1; ::_thesis: ( x0 = x1 & X1 is SubSpace of X0 & g is_continuous_at x0 implies g | X1 is_continuous_at x1 )
assume A1: x0 = x1 ; ::_thesis: ( not X1 is SubSpace of X0 or not g is_continuous_at x0 or g | X1 is_continuous_at x1 )
assume A2: X1 is SubSpace of X0 ; ::_thesis: ( not g is_continuous_at x0 or g | X1 is_continuous_at x1 )
assume A3: g is_continuous_at x0 ; ::_thesis: g | X1 is_continuous_at x1
for G being Subset of Y st G is open & (g | X1) . x1 in G holds
ex H0 being Subset of X1 st
( H0 is open & x1 in H0 & (g | X1) .: H0 c= G )
proof
reconsider C = the carrier of X1 as Subset of X0 by A2, TSEP_1:1;
let G be Subset of Y; ::_thesis: ( G is open & (g | X1) . x1 in G implies ex H0 being Subset of X1 st
( H0 is open & x1 in H0 & (g | X1) .: H0 c= G ) )
assume that
A4: G is open and
A5: (g | X1) . x1 in G ; ::_thesis: ex H0 being Subset of X1 st
( H0 is open & x1 in H0 & (g | X1) .: H0 c= G )
g . x0 in G by A1, A2, A5, Th65;
then consider H being Subset of X0 such that
A6: ( H is open & x0 in H ) and
A7: g .: H c= G by A3, A4, Th43;
reconsider H0 = H /\ C as Subset of X1 by XBOOLE_1:17;
( g .: H0 c= (g .: H) /\ (g .: C) & (g .: H) /\ (g .: C) c= g .: H ) by RELAT_1:121, XBOOLE_1:17;
then g .: H0 c= g .: H by XBOOLE_1:1;
then A8: g .: H0 c= G by A7, XBOOLE_1:1;
take H0 ; ::_thesis: ( H0 is open & x1 in H0 & (g | X1) .: H0 c= G )
g | X1 = g | C by A2, Def5;
then ( H0 = H /\ ([#] X1) & (g | X1) .: H0 c= g .: H0 ) by RELAT_1:128;
hence ( H0 is open & x1 in H0 & (g | X1) .: H0 c= G ) by A1, A2, A6, A8, TOPS_2:24, XBOOLE_0:def_4, XBOOLE_1:1; ::_thesis: verum
end;
hence g | X1 is_continuous_at x1 by Th43; ::_thesis: verum
end;
theorem Th75: :: TMAP_1:75
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for f being Function of X,Y st X1 is SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & f | X0 is_continuous_at x0 holds
f | X1 is_continuous_at x1
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for f being Function of X,Y st X1 is SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & f | X0 is_continuous_at x0 holds
f | X1 is_continuous_at x1
let X1, X0 be non empty SubSpace of X; ::_thesis: for f being Function of X,Y st X1 is SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & f | X0 is_continuous_at x0 holds
f | X1 is_continuous_at x1
let f be Function of X,Y; ::_thesis: ( X1 is SubSpace of X0 implies for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & f | X0 is_continuous_at x0 holds
f | X1 is_continuous_at x1 )
assume A1: X1 is SubSpace of X0 ; ::_thesis: for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & f | X0 is_continuous_at x0 holds
f | X1 is_continuous_at x1
let x0 be Point of X0; ::_thesis: for x1 being Point of X1 st x0 = x1 & f | X0 is_continuous_at x0 holds
f | X1 is_continuous_at x1
let x1 be Point of X1; ::_thesis: ( x0 = x1 & f | X0 is_continuous_at x0 implies f | X1 is_continuous_at x1 )
assume A2: x0 = x1 ; ::_thesis: ( not f | X0 is_continuous_at x0 or f | X1 is_continuous_at x1 )
assume f | X0 is_continuous_at x0 ; ::_thesis: f | X1 is_continuous_at x1
then (f | X0) | X1 is_continuous_at x1 by A1, A2, Th74;
hence f | X1 is_continuous_at x1 by A1, Th71; ::_thesis: verum
end;
theorem Th76: :: TMAP_1:76
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X0
for x0 being Point of X0
for x1 being Point of X1 st A c= the carrier of X1 & A is a_neighborhood of x0 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X0
for x0 being Point of X0
for x1 being Point of X1 st A c= the carrier of X1 & A is a_neighborhood of x0 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let X1, X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X0
for x0 being Point of X0
for x1 being Point of X1 st A c= the carrier of X1 & A is a_neighborhood of x0 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let g be Function of X0,Y; ::_thesis: ( X1 is SubSpace of X0 implies for A being Subset of X0
for x0 being Point of X0
for x1 being Point of X1 st A c= the carrier of X1 & A is a_neighborhood of x0 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume A1: X1 is SubSpace of X0 ; ::_thesis: for A being Subset of X0
for x0 being Point of X0
for x1 being Point of X1 st A c= the carrier of X1 & A is a_neighborhood of x0 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let A be Subset of X0; ::_thesis: for x0 being Point of X0
for x1 being Point of X1 st A c= the carrier of X1 & A is a_neighborhood of x0 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let x0 be Point of X0; ::_thesis: for x1 being Point of X1 st A c= the carrier of X1 & A is a_neighborhood of x0 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let x1 be Point of X1; ::_thesis: ( A c= the carrier of X1 & A is a_neighborhood of x0 & x0 = x1 implies ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume that
A2: A c= the carrier of X1 and
A3: A is a_neighborhood of x0 and
A4: x0 = x1 ; ::_thesis: ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
thus ( g is_continuous_at x0 implies g | X1 is_continuous_at x1 ) by A1, A4, Th74; ::_thesis: ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 )
thus ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 ) ::_thesis: verum
proof
assume A5: g | X1 is_continuous_at x1 ; ::_thesis: g is_continuous_at x0
for G being Subset of Y st G is open & g . x0 in G holds
ex H being Subset of X0 st
( H is open & x0 in H & g .: H c= G )
proof
let G be Subset of Y; ::_thesis: ( G is open & g . x0 in G implies ex H being Subset of X0 st
( H is open & x0 in H & g .: H c= G ) )
assume that
A6: G is open and
A7: g . x0 in G ; ::_thesis: ex H being Subset of X0 st
( H is open & x0 in H & g .: H c= G )
(g | X1) . x1 in G by A1, A4, A7, Th65;
then consider H1 being Subset of X1 such that
A8: H1 is open and
A9: x1 in H1 and
A10: (g | X1) .: H1 c= G by A5, A6, Th43;
consider V being Subset of X0 such that
A11: V is open and
A12: V c= A and
A13: x0 in V by A3, CONNSP_2:6;
reconsider V1 = V as Subset of X1 by A2, A12, XBOOLE_1:1;
A14: H1 /\ V1 c= V by XBOOLE_1:17;
then reconsider H = H1 /\ V1 as Subset of X0 by XBOOLE_1:1;
A15: for z being Point of Y st z in g .: H holds
z in G
proof
set f = g | X1;
let z be Point of Y; ::_thesis: ( z in g .: H implies z in G )
assume z in g .: H ; ::_thesis: z in G
then consider y being Point of X0 such that
A16: y in H and
A17: z = g . y by FUNCT_2:65;
y in V by A14, A16;
then y in A by A12;
then A18: z = (g | X1) . y by A1, A2, A17, Th65;
H1 /\ V1 c= H1 by XBOOLE_1:17;
then z in (g | X1) .: H1 by A16, A18, FUNCT_2:35;
hence z in G by A10; ::_thesis: verum
end;
take H ; ::_thesis: ( H is open & x0 in H & g .: H c= G )
V1 is open by A1, A11, TOPS_2:25;
then H1 /\ V1 is open by A8;
hence ( H is open & x0 in H & g .: H c= G ) by A1, A4, A9, A11, A13, A14, A15, SUBSET_1:2, TSEP_1:9, XBOOLE_0:def_4; ::_thesis: verum
end;
hence g is_continuous_at x0 by Th43; ::_thesis: verum
end;
end;
theorem Th77: :: TMAP_1:77
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X0
for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X0
for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let X1, X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X0
for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let g be Function of X0,Y; ::_thesis: ( X1 is SubSpace of X0 implies for A being Subset of X0
for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume A1: X1 is SubSpace of X0 ; ::_thesis: for A being Subset of X0
for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let A be Subset of X0; ::_thesis: for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let x0 be Point of X0; ::_thesis: for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let x1 be Point of X1; ::_thesis: ( A is open & x0 in A & A c= the carrier of X1 & x0 = x1 implies ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume that
A2: ( A is open & x0 in A ) and
A3: A c= the carrier of X1 and
A4: x0 = x1 ; ::_thesis: ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
thus ( g is_continuous_at x0 implies g | X1 is_continuous_at x1 ) by A1, A4, Th74; ::_thesis: ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 )
thus ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 ) ::_thesis: verum
proof
assume A5: g | X1 is_continuous_at x1 ; ::_thesis: g is_continuous_at x0
A is a_neighborhood of x0 by A2, CONNSP_2:3;
hence g is_continuous_at x0 by A1, A3, A4, A5, Th76; ::_thesis: verum
end;
end;
theorem :: TMAP_1:78
for Y, X being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X
for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
proof
let Y, X be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X
for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let X1, X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y st X1 is SubSpace of X0 holds
for A being Subset of X
for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let g be Function of X0,Y; ::_thesis: ( X1 is SubSpace of X0 implies for A being Subset of X
for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume A1: X1 is SubSpace of X0 ; ::_thesis: for A being Subset of X
for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let A be Subset of X; ::_thesis: for x0 being Point of X0
for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let x0 be Point of X0; ::_thesis: for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let x1 be Point of X1; ::_thesis: ( A is open & x0 in A & A c= the carrier of X1 & x0 = x1 implies ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume that
A2: A is open and
A3: x0 in A and
A4: A c= the carrier of X1 and
A5: x0 = x1 ; ::_thesis: ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
thus ( g is_continuous_at x0 implies g | X1 is_continuous_at x1 ) by A1, A5, Th74; ::_thesis: ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 )
thus ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 ) ::_thesis: verum
proof
the carrier of X1 c= the carrier of X0 by A1, TSEP_1:4;
then reconsider B = A as Subset of X0 by A4, XBOOLE_1:1;
assume A6: g | X1 is_continuous_at x1 ; ::_thesis: g is_continuous_at x0
B is open by A2, TOPS_2:25;
hence g is_continuous_at x0 by A1, A3, A4, A5, A6, Th77; ::_thesis: verum
end;
end;
theorem Th79: :: TMAP_1:79
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is open SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is open SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let X1, X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y st X1 is open SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let g be Function of X0,Y; ::_thesis: ( X1 is open SubSpace of X0 implies for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume A1: X1 is open SubSpace of X0 ; ::_thesis: for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let x0 be Point of X0; ::_thesis: for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let x1 be Point of X1; ::_thesis: ( x0 = x1 implies ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume A2: x0 = x1 ; ::_thesis: ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
hence ( g is_continuous_at x0 implies g | X1 is_continuous_at x1 ) by A1, Th74; ::_thesis: ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 )
thus ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 ) ::_thesis: verum
proof
reconsider A = the carrier of X1 as Subset of X0 by A1, TSEP_1:1;
assume A3: g | X1 is_continuous_at x1 ; ::_thesis: g is_continuous_at x0
A is open by A1, TSEP_1:16;
hence g is_continuous_at x0 by A1, A2, A3, Th77; ::_thesis: verum
end;
end;
theorem :: TMAP_1:80
for Y, X being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is open SubSpace of X & X1 is SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
proof
let Y, X be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st X1 is open SubSpace of X & X1 is SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let X1, X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y st X1 is open SubSpace of X & X1 is SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let g be Function of X0,Y; ::_thesis: ( X1 is open SubSpace of X & X1 is SubSpace of X0 implies for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume A1: X1 is open SubSpace of X ; ::_thesis: ( not X1 is SubSpace of X0 or for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume A2: X1 is SubSpace of X0 ; ::_thesis: for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let x0 be Point of X0; ::_thesis: for x1 being Point of X1 st x0 = x1 holds
( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
let x1 be Point of X1; ::_thesis: ( x0 = x1 implies ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )
assume A3: x0 = x1 ; ::_thesis: ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )
hence ( g is_continuous_at x0 implies g | X1 is_continuous_at x1 ) by A2, Th74; ::_thesis: ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 )
thus ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 ) ::_thesis: verum
proof
the carrier of X1 c= the carrier of X0 by A2, TSEP_1:4;
then A4: X1 is open SubSpace of X0 by A1, TSEP_1:19;
assume g | X1 is_continuous_at x1 ; ::_thesis: g is_continuous_at x0
hence g is_continuous_at x0 by A3, A4, Th79; ::_thesis: verum
end;
end;
theorem Th81: :: TMAP_1:81
for X, Y being non empty TopSpace
for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st TopStruct(# the carrier of X1, the topology of X1 #) = X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0 being non empty SubSpace of X
for g being Function of X0,Y st TopStruct(# the carrier of X1, the topology of X1 #) = X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0
let X1, X0 be non empty SubSpace of X; ::_thesis: for g being Function of X0,Y st TopStruct(# the carrier of X1, the topology of X1 #) = X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0
let g be Function of X0,Y; ::_thesis: ( TopStruct(# the carrier of X1, the topology of X1 #) = X0 implies for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0 )
reconsider Y1 = TopStruct(# the carrier of X1, the topology of X1 #) as TopSpace ;
assume A1: TopStruct(# the carrier of X1, the topology of X1 #) = X0 ; ::_thesis: for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0
then the carrier of X1 c= the carrier of X0 ;
then reconsider A = the carrier of X1 as Subset of X0 ;
A = [#] X0 by A1;
then A2: A is open ;
Y1 is SubSpace of X0 by A1, TSEP_1:2;
then A3: X1 is open SubSpace of X0 by A2, Th7, TSEP_1:16;
let x0 be Point of X0; ::_thesis: for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0
let x1 be Point of X1; ::_thesis: ( x0 = x1 & g | X1 is_continuous_at x1 implies g is_continuous_at x0 )
assume A4: x0 = x1 ; ::_thesis: ( not g | X1 is_continuous_at x1 or g is_continuous_at x0 )
assume g | X1 is_continuous_at x1 ; ::_thesis: g is_continuous_at x0
hence g is_continuous_at x0 by A4, A3, Th79; ::_thesis: verum
end;
theorem Th82: :: TMAP_1:82
for X, Y being non empty TopSpace
for X0, X1 being non empty SubSpace of X
for g being continuous Function of X0,Y st X1 is SubSpace of X0 holds
g | X1 is continuous Function of X1,Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X0, X1 being non empty SubSpace of X
for g being continuous Function of X0,Y st X1 is SubSpace of X0 holds
g | X1 is continuous Function of X1,Y
let X0, X1 be non empty SubSpace of X; ::_thesis: for g being continuous Function of X0,Y st X1 is SubSpace of X0 holds
g | X1 is continuous Function of X1,Y
let g be continuous Function of X0,Y; ::_thesis: ( X1 is SubSpace of X0 implies g | X1 is continuous Function of X1,Y )
assume A1: X1 is SubSpace of X0 ; ::_thesis: g | X1 is continuous Function of X1,Y
for x1 being Point of X1 holds g | X1 is_continuous_at x1
proof
let x1 be Point of X1; ::_thesis: g | X1 is_continuous_at x1
consider x0 being Point of X0 such that
A2: x0 = x1 by A1, Th10;
g is_continuous_at x0 by Th44;
hence g | X1 is_continuous_at x1 by A1, A2, Th74; ::_thesis: verum
end;
hence g | X1 is continuous Function of X1,Y by Th44; ::_thesis: verum
end;
theorem Th83: :: TMAP_1:83
for X, Y being non empty TopSpace
for X1, X0, X2 being non empty SubSpace of X st X1 is SubSpace of X0 & X2 is SubSpace of X1 holds
for g being Function of X0,Y st g | X1 is continuous Function of X1,Y holds
g | X2 is continuous Function of X2,Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X0, X2 being non empty SubSpace of X st X1 is SubSpace of X0 & X2 is SubSpace of X1 holds
for g being Function of X0,Y st g | X1 is continuous Function of X1,Y holds
g | X2 is continuous Function of X2,Y
let X1, X0, X2 be non empty SubSpace of X; ::_thesis: ( X1 is SubSpace of X0 & X2 is SubSpace of X1 implies for g being Function of X0,Y st g | X1 is continuous Function of X1,Y holds
g | X2 is continuous Function of X2,Y )
assume A1: X1 is SubSpace of X0 ; ::_thesis: ( not X2 is SubSpace of X1 or for g being Function of X0,Y st g | X1 is continuous Function of X1,Y holds
g | X2 is continuous Function of X2,Y )
assume A2: X2 is SubSpace of X1 ; ::_thesis: for g being Function of X0,Y st g | X1 is continuous Function of X1,Y holds
g | X2 is continuous Function of X2,Y
let g be Function of X0,Y; ::_thesis: ( g | X1 is continuous Function of X1,Y implies g | X2 is continuous Function of X2,Y )
assume g | X1 is continuous Function of X1,Y ; ::_thesis: g | X2 is continuous Function of X2,Y
then (g | X1) | X2 is continuous Function of X2,Y by A2, Th82;
hence g | X2 is continuous Function of X2,Y by A1, A2, Th72; ::_thesis: verum
end;
registration
let X be non empty TopSpace;
cluster id X -> continuous ;
coherence
id X is continuous
proof
( [#] X <> {} & ( for V being Subset of X st V is open holds
(id X) " V is open ) ) by FUNCT_2:94;
hence id X is continuous by TOPS_2:43; ::_thesis: verum
end;
end;
definition
let X be non empty TopSpace;
let X0 be SubSpace of X;
func incl X0 -> Function of X0,X equals :: TMAP_1:def 6
(id X) | X0;
coherence
(id X) | X0 is Function of X0,X ;
correctness
;
end;
:: deftheorem defines incl TMAP_1:def_6_:_
for X being non empty TopSpace
for X0 being SubSpace of X holds incl X0 = (id X) | X0;
notation
let X be non empty TopSpace;
let X0 be SubSpace of X;
synonym X0 incl X for incl X0;
end;
theorem :: TMAP_1:84
for X being non empty TopSpace
for X0 being non empty SubSpace of X
for x being Point of X st x in the carrier of X0 holds
(incl X0) . x = x
proof
let X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for x being Point of X st x in the carrier of X0 holds
(incl X0) . x = x
let X0 be non empty SubSpace of X; ::_thesis: for x being Point of X st x in the carrier of X0 holds
(incl X0) . x = x
let x be Point of X; ::_thesis: ( x in the carrier of X0 implies (incl X0) . x = x )
assume x in the carrier of X0 ; ::_thesis: (incl X0) . x = x
hence (incl X0) . x = (id X) . x by FUNCT_1:49
.= x by FUNCT_1:18 ;
::_thesis: verum
end;
theorem :: TMAP_1:85
for X being non empty TopSpace
for X0 being non empty SubSpace of X
for f0 being Function of X0,X st ( for x being Point of X st x in the carrier of X0 holds
x = f0 . x ) holds
incl X0 = f0
proof
let X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for f0 being Function of X0,X st ( for x being Point of X st x in the carrier of X0 holds
x = f0 . x ) holds
incl X0 = f0
let X0 be non empty SubSpace of X; ::_thesis: for f0 being Function of X0,X st ( for x being Point of X st x in the carrier of X0 holds
x = f0 . x ) holds
incl X0 = f0
let f0 be Function of X0,X; ::_thesis: ( ( for x being Point of X st x in the carrier of X0 holds
x = f0 . x ) implies incl X0 = f0 )
assume A1: for x being Point of X st x in the carrier of X0 holds
x = f0 . x ; ::_thesis: incl X0 = f0
now__::_thesis:_for_x_being_Point_of_X_st_x_in_the_carrier_of_X0_holds_
(id_X)_._x_=_f0_._x
let x be Point of X; ::_thesis: ( x in the carrier of X0 implies (id X) . x = f0 . x )
assume A2: x in the carrier of X0 ; ::_thesis: (id X) . x = f0 . x
(id X) . x = x by FUNCT_1:18;
hence (id X) . x = f0 . x by A1, A2; ::_thesis: verum
end;
hence incl X0 = f0 by Th53; ::_thesis: verum
end;
theorem :: TMAP_1:86
for X, Y being non empty TopSpace
for X0 being non empty SubSpace of X
for f being Function of X,Y holds f | X0 = f * (incl X0)
proof
let X, Y be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for f being Function of X,Y holds f | X0 = f * (incl X0)
let X0 be non empty SubSpace of X; ::_thesis: for f being Function of X,Y holds f | X0 = f * (incl X0)
let f be Function of X,Y; ::_thesis: f | X0 = f * (incl X0)
thus f | X0 = (f * (id X)) | X0 by FUNCT_2:17
.= f * (incl X0) by Th62 ; ::_thesis: verum
end;
theorem :: TMAP_1:87
for X being non empty TopSpace
for X0 being non empty SubSpace of X holds incl X0 is continuous Function of X0,X ;
begin
definition
let X be non empty TopSpace;
let A be Subset of X;
funcA -extension_of_the_topology_of X -> Subset-Family of X equals :: TMAP_1:def 7
{ (H \/ (G /\ A)) where H, G is Subset of X : ( H in the topology of X & G in the topology of X ) } ;
coherence
{ (H \/ (G /\ A)) where H, G is Subset of X : ( H in the topology of X & G in the topology of X ) } is Subset-Family of X
proof
set FF = { (H \/ (G /\ A)) where H, G is Subset of X : ( H in the topology of X & G in the topology of X ) } ;
now__::_thesis:_for_C_being_set_st_C_in__{__(H_\/_(G_/\_A))_where_H,_G_is_Subset_of_X_:_(_H_in_the_topology_of_X_&_G_in_the_topology_of_X_)__}__holds_
C_in_bool_the_carrier_of_X
let C be set ; ::_thesis: ( C in { (H \/ (G /\ A)) where H, G is Subset of X : ( H in the topology of X & G in the topology of X ) } implies C in bool the carrier of X )
assume C in { (H \/ (G /\ A)) where H, G is Subset of X : ( H in the topology of X & G in the topology of X ) } ; ::_thesis: C in bool the carrier of X
then ex P, S being Subset of X st
( C = P \/ (S /\ A) & P in the topology of X & S in the topology of X ) ;
hence C in bool the carrier of X ; ::_thesis: verum
end;
hence { (H \/ (G /\ A)) where H, G is Subset of X : ( H in the topology of X & G in the topology of X ) } is Subset-Family of X by TARSKI:def_3; ::_thesis: verum
end;
end;
:: deftheorem defines -extension_of_the_topology_of TMAP_1:def_7_:_
for X being non empty TopSpace
for A being Subset of X holds A -extension_of_the_topology_of X = { (H \/ (G /\ A)) where H, G is Subset of X : ( H in the topology of X & G in the topology of X ) } ;
theorem Th88: :: TMAP_1:88
for X being non empty TopSpace
for A being Subset of X holds the topology of X c= A -extension_of_the_topology_of X
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X holds the topology of X c= A -extension_of_the_topology_of X
let A be Subset of X; ::_thesis: the topology of X c= A -extension_of_the_topology_of X
now__::_thesis:_for_W_being_set_st_W_in_the_topology_of_X_holds_
W_in_A_-extension_of_the_topology_of_X
{} X = {} ;
then reconsider G = {} as Subset of X ;
let W be set ; ::_thesis: ( W in the topology of X implies W in A -extension_of_the_topology_of X )
assume A1: W in the topology of X ; ::_thesis: W in A -extension_of_the_topology_of X
then reconsider H = W as Subset of X ;
( H = H \/ (G /\ A) & G in the topology of X ) by PRE_TOPC:1;
hence W in A -extension_of_the_topology_of X by A1; ::_thesis: verum
end;
hence the topology of X c= A -extension_of_the_topology_of X by TARSKI:def_3; ::_thesis: verum
end;
theorem Th89: :: TMAP_1:89
for X being non empty TopSpace
for A being Subset of X holds { (G /\ A) where G is Subset of X : G in the topology of X } c= A -extension_of_the_topology_of X
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X holds { (G /\ A) where G is Subset of X : G in the topology of X } c= A -extension_of_the_topology_of X
let A be Subset of X; ::_thesis: { (G /\ A) where G is Subset of X : G in the topology of X } c= A -extension_of_the_topology_of X
now__::_thesis:_for_W_being_set_st_W_in__{__(G_/\_A)_where_G_is_Subset_of_X_:_G_in_the_topology_of_X__}__holds_
W_in_A_-extension_of_the_topology_of_X
{} X = {} ;
then reconsider H = {} as Subset of X ;
let W be set ; ::_thesis: ( W in { (G /\ A) where G is Subset of X : G in the topology of X } implies W in A -extension_of_the_topology_of X )
assume W in { (G /\ A) where G is Subset of X : G in the topology of X } ; ::_thesis: W in A -extension_of_the_topology_of X
then consider G being Subset of X such that
A1: ( W = G /\ A & G in the topology of X ) ;
( G /\ A = H \/ (G /\ A) & H in the topology of X ) by PRE_TOPC:1;
hence W in A -extension_of_the_topology_of X by A1; ::_thesis: verum
end;
hence { (G /\ A) where G is Subset of X : G in the topology of X } c= A -extension_of_the_topology_of X by TARSKI:def_3; ::_thesis: verum
end;
theorem Th90: :: TMAP_1:90
for X being non empty TopSpace
for A, C, D being Subset of X st C in the topology of X & D in { (G /\ A) where G is Subset of X : G in the topology of X } holds
( C \/ D in A -extension_of_the_topology_of X & C /\ D in A -extension_of_the_topology_of X )
proof
let X be non empty TopSpace; ::_thesis: for A, C, D being Subset of X st C in the topology of X & D in { (G /\ A) where G is Subset of X : G in the topology of X } holds
( C \/ D in A -extension_of_the_topology_of X & C /\ D in A -extension_of_the_topology_of X )
let A be Subset of X; ::_thesis: for C, D being Subset of X st C in the topology of X & D in { (G /\ A) where G is Subset of X : G in the topology of X } holds
( C \/ D in A -extension_of_the_topology_of X & C /\ D in A -extension_of_the_topology_of X )
let C, D be Subset of X; ::_thesis: ( C in the topology of X & D in { (G /\ A) where G is Subset of X : G in the topology of X } implies ( C \/ D in A -extension_of_the_topology_of X & C /\ D in A -extension_of_the_topology_of X ) )
assume A1: C in the topology of X ; ::_thesis: ( not D in { (G /\ A) where G is Subset of X : G in the topology of X } or ( C \/ D in A -extension_of_the_topology_of X & C /\ D in A -extension_of_the_topology_of X ) )
assume D in { (G /\ A) where G is Subset of X : G in the topology of X } ; ::_thesis: ( C \/ D in A -extension_of_the_topology_of X & C /\ D in A -extension_of_the_topology_of X )
then consider G being Subset of X such that
A2: D = G /\ A and
A3: G in the topology of X ;
thus C \/ D in A -extension_of_the_topology_of X by A1, A2, A3; ::_thesis: C /\ D in A -extension_of_the_topology_of X
thus C /\ D in A -extension_of_the_topology_of X ::_thesis: verum
proof
{} X = {} ;
then reconsider H = {} as Subset of X ;
A4: ( C /\ D = H \/ ((C /\ G) /\ A) & H in the topology of X ) by A2, PRE_TOPC:1, XBOOLE_1:16;
C /\ G in the topology of X by A1, A3, PRE_TOPC:def_1;
hence C /\ D in A -extension_of_the_topology_of X by A4; ::_thesis: verum
end;
end;
theorem Th91: :: TMAP_1:91
for X being non empty TopSpace
for A being Subset of X holds A in A -extension_of_the_topology_of X
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X holds A in A -extension_of_the_topology_of X
let A be Subset of X; ::_thesis: A in A -extension_of_the_topology_of X
X is SubSpace of X by TSEP_1:2;
then reconsider G = the carrier of X as Subset of X by TSEP_1:1;
A1: G in the topology of X by PRE_TOPC:def_1;
{} X = {} ;
then reconsider H = {} as Subset of X ;
( A = H \/ (G /\ A) & H in the topology of X ) by PRE_TOPC:1, XBOOLE_1:28;
hence A in A -extension_of_the_topology_of X by A1; ::_thesis: verum
end;
theorem Th92: :: TMAP_1:92
for X being non empty TopSpace
for A being Subset of X holds
( A in the topology of X iff the topology of X = A -extension_of_the_topology_of X )
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X holds
( A in the topology of X iff the topology of X = A -extension_of_the_topology_of X )
let A be Subset of X; ::_thesis: ( A in the topology of X iff the topology of X = A -extension_of_the_topology_of X )
thus ( A in the topology of X implies the topology of X = A -extension_of_the_topology_of X ) ::_thesis: ( the topology of X = A -extension_of_the_topology_of X implies A in the topology of X )
proof
assume A1: A in the topology of X ; ::_thesis: the topology of X = A -extension_of_the_topology_of X
now__::_thesis:_for_W_being_set_st_W_in_A_-extension_of_the_topology_of_X_holds_
W_in_the_topology_of_X
let W be set ; ::_thesis: ( W in A -extension_of_the_topology_of X implies W in the topology of X )
assume W in A -extension_of_the_topology_of X ; ::_thesis: W in the topology of X
then consider H, G being Subset of X such that
A2: W = H \/ (G /\ A) and
A3: H in the topology of X and
A4: G in the topology of X ;
reconsider H1 = H as Subset of X ;
G /\ A in the topology of X by A1, A4, PRE_TOPC:def_1;
then A5: G /\ A is open by PRE_TOPC:def_2;
H1 is open by A3, PRE_TOPC:def_2;
then H1 \/ (G /\ A) is open by A5;
hence W in the topology of X by A2, PRE_TOPC:def_2; ::_thesis: verum
end;
then A6: A -extension_of_the_topology_of X c= the topology of X by TARSKI:def_3;
the topology of X c= A -extension_of_the_topology_of X by Th88;
hence the topology of X = A -extension_of_the_topology_of X by A6, XBOOLE_0:def_10; ::_thesis: verum
end;
thus ( the topology of X = A -extension_of_the_topology_of X implies A in the topology of X ) by Th91; ::_thesis: verum
end;
definition
let X be non empty TopSpace;
let A be Subset of X;
funcX modified_with_respect_to A -> strict TopSpace equals :: TMAP_1:def 8
TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #);
coherence
TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) is strict TopSpace
proof
set Y = TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #);
A1: for C, D being Subset of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) st C in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) & D in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) holds
C /\ D in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #)
proof
let C, D be Subset of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #); ::_thesis: ( C in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) & D in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) implies C /\ D in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) )
assume that
A2: C in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) and
A3: D in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) ; ::_thesis: C /\ D in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #)
consider H1, G1 being Subset of X such that
A4: C = H1 \/ (G1 /\ A) and
A5: H1 in the topology of X and
A6: G1 in the topology of X by A2;
consider H2, G2 being Subset of X such that
A7: D = H2 \/ (G2 /\ A) and
A8: H2 in the topology of X and
A9: G2 in the topology of X by A3;
A10: C /\ D = (H1 /\ (H2 \/ (G2 /\ A))) \/ ((G1 /\ A) /\ (H2 \/ (G2 /\ A))) by A4, A7, XBOOLE_1:23
.= ((H1 /\ H2) \/ (H1 /\ (G2 /\ A))) \/ ((G1 /\ A) /\ (H2 \/ (G2 /\ A))) by XBOOLE_1:23
.= ((H1 /\ H2) \/ (H1 /\ (G2 /\ A))) \/ (((G1 /\ A) /\ H2) \/ ((G1 /\ A) /\ (G2 /\ A))) by XBOOLE_1:23
.= ((H1 /\ H2) \/ ((H1 /\ G2) /\ A)) \/ (((G1 /\ A) /\ H2) \/ ((G1 /\ A) /\ (G2 /\ A))) by XBOOLE_1:16
.= ((H1 /\ H2) \/ ((H1 /\ G2) /\ A)) \/ (((G1 /\ A) /\ H2) \/ (G1 /\ ((G2 /\ A) /\ A))) by XBOOLE_1:16
.= ((H1 /\ H2) \/ ((H1 /\ G2) /\ A)) \/ (((G1 /\ A) /\ H2) \/ (G1 /\ (G2 /\ (A /\ A)))) by XBOOLE_1:16
.= ((H1 /\ H2) \/ ((H1 /\ G2) /\ A)) \/ ((H2 /\ (G1 /\ A)) \/ ((G1 /\ G2) /\ A)) by XBOOLE_1:16
.= ((H1 /\ H2) \/ ((H1 /\ G2) /\ A)) \/ (((H2 /\ G1) /\ A) \/ ((G1 /\ G2) /\ A)) by XBOOLE_1:16
.= ((H1 /\ H2) \/ ((H1 /\ G2) /\ A)) \/ (((H2 /\ G1) \/ (G1 /\ G2)) /\ A) by XBOOLE_1:23
.= (H1 /\ H2) \/ (((H1 /\ G2) /\ A) \/ (((H2 /\ G1) \/ (G1 /\ G2)) /\ A)) by XBOOLE_1:4
.= (H1 /\ H2) \/ (((H1 /\ G2) \/ ((H2 /\ G1) \/ (G1 /\ G2))) /\ A) by XBOOLE_1:23
.= (H1 /\ H2) \/ ((((H1 /\ G2) \/ (H2 /\ G1)) \/ (G1 /\ G2)) /\ A) by XBOOLE_1:4 ;
G1 /\ G2 in the topology of X by A6, A9, PRE_TOPC:def_1;
then A11: G1 /\ G2 is open by PRE_TOPC:def_2;
H2 /\ G1 in the topology of X by A6, A8, PRE_TOPC:def_1;
then A12: H2 /\ G1 is open by PRE_TOPC:def_2;
H1 /\ G2 in the topology of X by A5, A9, PRE_TOPC:def_1;
then H1 /\ G2 is open by PRE_TOPC:def_2;
then (H1 /\ G2) \/ (H2 /\ G1) is open by A12;
then ((H1 /\ G2) \/ (H2 /\ G1)) \/ (G1 /\ G2) is open by A11;
then A13: ((H1 /\ G2) \/ (H2 /\ G1)) \/ (G1 /\ G2) in the topology of X by PRE_TOPC:def_2;
H1 /\ H2 in the topology of X by A5, A8, PRE_TOPC:def_1;
hence C /\ D in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) by A13, A10; ::_thesis: verum
end;
A14: for FF being Subset-Family of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) st FF c= the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) holds
union FF in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #)
proof
set SAA = { (G /\ A) where G is Subset of X : G in the topology of X } ;
{ (G /\ A) where G is Subset of X : G in the topology of X } c= A -extension_of_the_topology_of X by Th89;
then reconsider SAA = { (G /\ A) where G is Subset of X : G in the topology of X } as Subset-Family of X by TOPS_2:2;
let FF be Subset-Family of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #); ::_thesis: ( FF c= the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) implies union FF in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) )
set AA = { (P \/ (S /\ A)) where P, S is Subset of X : ( P in the topology of X & S in the topology of X ) } ;
set FF1 = { H where H is Subset of X : ( H in the topology of X & ex G being Subset of X st
( G in the topology of X & H \/ (G /\ A) in FF ) ) } ;
set FF2 = { (G /\ A) where G is Subset of X : ( G in the topology of X & ex H being Subset of X st
( H in the topology of X & H \/ (G /\ A) in FF ) ) } ;
now__::_thesis:_for_W_being_set_st_W_in__{__H_where_H_is_Subset_of_X_:_(_H_in_the_topology_of_X_&_ex_G_being_Subset_of_X_st_
(_G_in_the_topology_of_X_&_H_\/_(G_/\_A)_in_FF_)_)__}__holds_
W_in_the_topology_of_X
let W be set ; ::_thesis: ( W in { H where H is Subset of X : ( H in the topology of X & ex G being Subset of X st
( G in the topology of X & H \/ (G /\ A) in FF ) ) } implies W in the topology of X )
assume W in { H where H is Subset of X : ( H in the topology of X & ex G being Subset of X st
( G in the topology of X & H \/ (G /\ A) in FF ) ) } ; ::_thesis: W in the topology of X
then ex H being Subset of X st
( W = H & H in the topology of X & ex G being Subset of X st
( G in the topology of X & H \/ (G /\ A) in FF ) ) ;
hence W in the topology of X ; ::_thesis: verum
end;
then A15: { H where H is Subset of X : ( H in the topology of X & ex G being Subset of X st
( G in the topology of X & H \/ (G /\ A) in FF ) ) } c= the topology of X by TARSKI:def_3;
now__::_thesis:_for_W_being_set_st_W_in__{__(G_/\_A)_where_G_is_Subset_of_X_:_(_G_in_the_topology_of_X_&_ex_H_being_Subset_of_X_st_
(_H_in_the_topology_of_X_&_H_\/_(G_/\_A)_in_FF_)_)__}__holds_
W_in_SAA
let W be set ; ::_thesis: ( W in { (G /\ A) where G is Subset of X : ( G in the topology of X & ex H being Subset of X st
( H in the topology of X & H \/ (G /\ A) in FF ) ) } implies W in SAA )
assume W in { (G /\ A) where G is Subset of X : ( G in the topology of X & ex H being Subset of X st
( H in the topology of X & H \/ (G /\ A) in FF ) ) } ; ::_thesis: W in SAA
then ex G being Subset of X st
( W = G /\ A & G in the topology of X & ex H being Subset of X st
( H in the topology of X & H \/ (G /\ A) in FF ) ) ;
hence W in SAA ; ::_thesis: verum
end;
then A16: { (G /\ A) where G is Subset of X : ( G in the topology of X & ex H being Subset of X st
( H in the topology of X & H \/ (G /\ A) in FF ) ) } c= SAA by TARSKI:def_3;
then reconsider FF2 = { (G /\ A) where G is Subset of X : ( G in the topology of X & ex H being Subset of X st
( H in the topology of X & H \/ (G /\ A) in FF ) ) } as Subset-Family of X by TOPS_2:2;
A17: union FF2 in SAA
proof
now__::_thesis:_union_FF2_in_SAA
percases ( A = {} or A <> {} ) ;
supposeA18: A = {} ; ::_thesis: union FF2 in SAA
now__::_thesis:_for_W_being_set_st_W_in_{{}}_holds_
W_in_SAA
let W be set ; ::_thesis: ( W in {{}} implies W in SAA )
assume W in {{}} ; ::_thesis: W in SAA
then A19: W = {} /\ A by TARSKI:def_1;
{} in the topology of X by PRE_TOPC:1;
hence W in SAA by A19; ::_thesis: verum
end;
then A20: {{}} c= SAA by TARSKI:def_3;
now__::_thesis:_for_W_being_set_st_W_in_SAA_holds_
W_in_{{}}
let W be set ; ::_thesis: ( W in SAA implies W in {{}} )
assume W in SAA ; ::_thesis: W in {{}}
then ex G being Subset of X st
( W = G /\ A & G in the topology of X ) ;
hence W in {{}} by A18, TARSKI:def_1; ::_thesis: verum
end;
then SAA c= {{}} by TARSKI:def_3;
then A21: SAA = {{}} by A20, XBOOLE_0:def_10;
now__::_thesis:_union_FF2_=_{}
percases ( FF2 = {{}} or FF2 = {} ) by A16, A21, ZFMISC_1:33;
suppose FF2 = {{}} ; ::_thesis: union FF2 = {}
hence union FF2 = {} by ZFMISC_1:25; ::_thesis: verum
end;
suppose FF2 = {} ; ::_thesis: union FF2 = {}
hence union FF2 = {} by ZFMISC_1:2; ::_thesis: verum
end;
end;
end;
hence union FF2 in SAA by A21, TARSKI:def_1; ::_thesis: verum
end;
suppose A <> {} ; ::_thesis: union FF2 in SAA
then consider Y being non empty strict SubSpace of X such that
A22: A = the carrier of Y by TSEP_1:10;
now__::_thesis:_for_W_being_set_st_W_in_SAA_holds_
W_in_the_topology_of_Y
let W be set ; ::_thesis: ( W in SAA implies W in the topology of Y )
assume W in SAA ; ::_thesis: W in the topology of Y
then consider G being Subset of X such that
A23: W = G /\ A and
A24: G in the topology of X ;
reconsider C = G /\ ([#] Y) as Subset of Y ;
C in the topology of Y by A24, PRE_TOPC:def_4;
hence W in the topology of Y by A22, A23; ::_thesis: verum
end;
then A25: SAA c= the topology of Y by TARSKI:def_3;
A26: now__::_thesis:_for_W_being_set_st_W_in_the_topology_of_Y_holds_
W_in_SAA
let W be set ; ::_thesis: ( W in the topology of Y implies W in SAA )
assume A27: W in the topology of Y ; ::_thesis: W in SAA
then reconsider C = W as Subset of Y ;
ex G being Subset of X st
( G in the topology of X & C = G /\ ([#] Y) ) by A27, PRE_TOPC:def_4;
hence W in SAA by A22; ::_thesis: verum
end;
then the topology of Y c= SAA by TARSKI:def_3;
then A28: the topology of Y = SAA by A25, XBOOLE_0:def_10;
then reconsider SS = FF2 as Subset-Family of Y by A16, TOPS_2:2;
union SS in the topology of Y by A16, A28, PRE_TOPC:def_1;
hence union FF2 in SAA by A26; ::_thesis: verum
end;
end;
end;
hence union FF2 in SAA ; ::_thesis: verum
end;
reconsider FF1 = { H where H is Subset of X : ( H in the topology of X & ex G being Subset of X st
( G in the topology of X & H \/ (G /\ A) in FF ) ) } as Subset-Family of X by A15, TOPS_2:2;
A29: union FF1 in the topology of X by A15, PRE_TOPC:def_1;
now__::_thesis:_for_x_being_set_st_x_in_(union_FF1)_\/_(union_FF2)_holds_
x_in_union_FF
let x be set ; ::_thesis: ( x in (union FF1) \/ (union FF2) implies x in union FF )
assume A30: x in (union FF1) \/ (union FF2) ; ::_thesis: x in union FF
now__::_thesis:_x_in_union_FF
percases ( x in union FF1 or x in union FF2 ) by A30, XBOOLE_0:def_3;
suppose x in union FF1 ; ::_thesis: x in union FF
then consider W being set such that
A31: x in W and
A32: W in FF1 by TARSKI:def_4;
consider H being Subset of X such that
A33: W = H and
H in the topology of X and
A34: ex G being Subset of X st
( G in the topology of X & H \/ (G /\ A) in FF ) by A32;
consider G being Subset of X such that
A35: H \/ (G /\ A) in FF by A34;
A36: H c= H \/ (G /\ A) by XBOOLE_1:7;
H \/ (G /\ A) c= union FF by A35, ZFMISC_1:74;
then H c= union FF by A36, XBOOLE_1:1;
hence x in union FF by A31, A33; ::_thesis: verum
end;
suppose x in union FF2 ; ::_thesis: x in union FF
then consider W being set such that
A37: x in W and
A38: W in FF2 by TARSKI:def_4;
consider G being Subset of X such that
A39: W = G /\ A and
G in the topology of X and
A40: ex H being Subset of X st
( H in the topology of X & H \/ (G /\ A) in FF ) by A38;
consider H being Subset of X such that
A41: H \/ (G /\ A) in FF by A40;
A42: G /\ A c= H \/ (G /\ A) by XBOOLE_1:7;
H \/ (G /\ A) c= union FF by A41, ZFMISC_1:74;
then G /\ A c= union FF by A42, XBOOLE_1:1;
hence x in union FF by A37, A39; ::_thesis: verum
end;
end;
end;
hence x in union FF ; ::_thesis: verum
end;
then A43: (union FF1) \/ (union FF2) c= union FF by TARSKI:def_3;
assume A44: FF c= the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) ; ::_thesis: union FF in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #)
now__::_thesis:_for_x_being_set_st_x_in_union_FF_holds_
x_in_(union_FF1)_\/_(union_FF2)
let x be set ; ::_thesis: ( x in union FF implies x in (union FF1) \/ (union FF2) )
A45: union FF1 c= (union FF1) \/ (union FF2) by XBOOLE_1:7;
A46: union FF2 c= (union FF1) \/ (union FF2) by XBOOLE_1:7;
assume x in union FF ; ::_thesis: x in (union FF1) \/ (union FF2)
then consider W being set such that
A47: x in W and
A48: W in FF by TARSKI:def_4;
W in { (P \/ (S /\ A)) where P, S is Subset of X : ( P in the topology of X & S in the topology of X ) } by A44, A48;
then consider H, G being Subset of X such that
A49: W = H \/ (G /\ A) and
A50: ( H in the topology of X & G in the topology of X ) ;
G /\ A in FF2 by A48, A49, A50;
then G /\ A c= union FF2 by ZFMISC_1:74;
then A51: G /\ A c= (union FF1) \/ (union FF2) by A46, XBOOLE_1:1;
H in FF1 by A48, A49, A50;
then H c= union FF1 by ZFMISC_1:74;
then H c= (union FF1) \/ (union FF2) by A45, XBOOLE_1:1;
then H \/ (G /\ A) c= (union FF1) \/ (union FF2) by A51, XBOOLE_1:8;
hence x in (union FF1) \/ (union FF2) by A47, A49; ::_thesis: verum
end;
then union FF c= (union FF1) \/ (union FF2) by TARSKI:def_3;
then union FF = (union FF1) \/ (union FF2) by A43, XBOOLE_0:def_10;
hence union FF in the topology of TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) by A29, A17, Th90; ::_thesis: verum
end;
( the topology of X c= A -extension_of_the_topology_of X & the carrier of X in the topology of X ) by Th88, PRE_TOPC:def_1;
hence TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #) is strict TopSpace by A14, A1, PRE_TOPC:def_1; ::_thesis: verum
end;
end;
:: deftheorem defines modified_with_respect_to TMAP_1:def_8_:_
for X being non empty TopSpace
for A being Subset of X holds X modified_with_respect_to A = TopStruct(# the carrier of X,(A -extension_of_the_topology_of X) #);
registration
let X be non empty TopSpace;
let A be Subset of X;
clusterX modified_with_respect_to A -> non empty strict ;
coherence
not X modified_with_respect_to A is empty ;
end;
theorem :: TMAP_1:93
for X being non empty TopSpace
for A being Subset of X holds
( the carrier of (X modified_with_respect_to A) = the carrier of X & the topology of (X modified_with_respect_to A) = A -extension_of_the_topology_of X ) ;
theorem Th94: :: TMAP_1:94
for X being non empty TopSpace
for A being Subset of X
for B being Subset of (X modified_with_respect_to A) st B = A holds
B is open
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X
for B being Subset of (X modified_with_respect_to A) st B = A holds
B is open
let A be Subset of X; ::_thesis: for B being Subset of (X modified_with_respect_to A) st B = A holds
B is open
let B be Subset of (X modified_with_respect_to A); ::_thesis: ( B = A implies B is open )
assume B = A ; ::_thesis: B is open
then B in A -extension_of_the_topology_of X by Th91;
hence B is open by PRE_TOPC:def_2; ::_thesis: verum
end;
theorem Th95: :: TMAP_1:95
for X being non empty TopSpace
for A being Subset of X holds
( A is open iff TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A )
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X holds
( A is open iff TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A )
let A be Subset of X; ::_thesis: ( A is open iff TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A )
thus ( A is open implies TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A ) ::_thesis: ( TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A implies A is open )
proof
assume A is open ; ::_thesis: TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A
then A in the topology of X by PRE_TOPC:def_2;
hence TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A by Th92; ::_thesis: verum
end;
thus ( TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A implies A is open ) ::_thesis: verum
proof
assume TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A ; ::_thesis: A is open
then A in the topology of X by Th92;
hence A is open by PRE_TOPC:def_2; ::_thesis: verum
end;
end;
definition
let X be non empty TopSpace;
let A be Subset of X;
func modid (X,A) -> Function of X,(X modified_with_respect_to A) equals :: TMAP_1:def 9
id the carrier of X;
coherence
id the carrier of X is Function of X,(X modified_with_respect_to A) ;
end;
:: deftheorem defines modid TMAP_1:def_9_:_
for X being non empty TopSpace
for A being Subset of X holds modid (X,A) = id the carrier of X;
theorem Th96: :: TMAP_1:96
for X being non empty TopSpace
for A being Subset of X
for x being Point of X st not x in A holds
modid (X,A) is_continuous_at x
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X
for x being Point of X st not x in A holds
modid (X,A) is_continuous_at x
let A be Subset of X; ::_thesis: for x being Point of X st not x in A holds
modid (X,A) is_continuous_at x
let x be Point of X; ::_thesis: ( not x in A implies modid (X,A) is_continuous_at x )
assume A1: not x in A ; ::_thesis: modid (X,A) is_continuous_at x
now__::_thesis:_for_W_being_Subset_of_(X_modified_with_respect_to_A)_st_W_is_open_&_(modid_(X,A))_._x_in_W_holds_
ex_V_being_Subset_of_X_st_
(_V_is_open_&_x_in_V_&_(modid_(X,A))_.:_V_c=_W_)
let W be Subset of (X modified_with_respect_to A); ::_thesis: ( W is open & (modid (X,A)) . x in W implies ex V being Subset of X st
( V is open & x in V & (modid (X,A)) .: V c= W ) )
assume that
A2: W is open and
A3: (modid (X,A)) . x in W ; ::_thesis: ex V being Subset of X st
( V is open & x in V & (modid (X,A)) .: V c= W )
W in A -extension_of_the_topology_of X by A2, PRE_TOPC:def_2;
then consider H, G being Subset of X such that
A4: W = H \/ (G /\ A) and
A5: H in the topology of X and
G in the topology of X ;
A6: G /\ A c= A by XBOOLE_1:17;
(modid (X,A)) . x = x by FUNCT_1:18;
then A7: ( x in H or x in G /\ A ) by A3, A4, XBOOLE_0:def_3;
thus ex V being Subset of X st
( V is open & x in V & (modid (X,A)) .: V c= W ) ::_thesis: verum
proof
reconsider H = H as Subset of X ;
take H ; ::_thesis: ( H is open & x in H & (modid (X,A)) .: H c= W )
(modid (X,A)) .: H = H by FUNCT_1:92;
hence ( H is open & x in H & (modid (X,A)) .: H c= W ) by A1, A4, A5, A7, A6, PRE_TOPC:def_2, XBOOLE_1:7; ::_thesis: verum
end;
end;
hence modid (X,A) is_continuous_at x by Th43; ::_thesis: verum
end;
theorem Th97: :: TMAP_1:97
for X being non empty TopSpace
for A being Subset of X
for X0 being non empty SubSpace of X st the carrier of X0 misses A holds
for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X
for X0 being non empty SubSpace of X st the carrier of X0 misses A holds
for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0
let A be Subset of X; ::_thesis: for X0 being non empty SubSpace of X st the carrier of X0 misses A holds
for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0
let X0 be non empty SubSpace of X; ::_thesis: ( the carrier of X0 misses A implies for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0 )
assume A1: the carrier of X0 /\ A = {} ; :: according to XBOOLE_0:def_7 ::_thesis: for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0
let x0 be Point of X0; ::_thesis: (modid (X,A)) | X0 is_continuous_at x0
( x0 in the carrier of X0 & the carrier of X0 c= the carrier of X ) by BORSUK_1:1;
then reconsider x = x0 as Point of X ;
not x in A by A1, XBOOLE_0:def_4;
hence (modid (X,A)) | X0 is_continuous_at x0 by Th58, Th96; ::_thesis: verum
end;
theorem Th98: :: TMAP_1:98
for X being non empty TopSpace
for A being Subset of X
for X0 being non empty SubSpace of X st the carrier of X0 = A holds
for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X
for X0 being non empty SubSpace of X st the carrier of X0 = A holds
for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0
let A be Subset of X; ::_thesis: for X0 being non empty SubSpace of X st the carrier of X0 = A holds
for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0
let X0 be non empty SubSpace of X; ::_thesis: ( the carrier of X0 = A implies for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0 )
assume A1: the carrier of X0 = A ; ::_thesis: for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0
let x0 be Point of X0; ::_thesis: (modid (X,A)) | X0 is_continuous_at x0
now__::_thesis:_for_W_being_Subset_of_(X_modified_with_respect_to_A)_st_W_is_open_&_((modid_(X,A))_|_X0)_._x0_in_W_holds_
ex_V_being_Subset_of_X0_st_
(_V_is_open_&_x0_in_V_&_((modid_(X,A))_|_X0)_.:_V_c=_W_)
( x0 in the carrier of X0 & the carrier of X0 c= the carrier of X ) by BORSUK_1:1;
then reconsider x = x0 as Point of X ;
let W be Subset of (X modified_with_respect_to A); ::_thesis: ( W is open & ((modid (X,A)) | X0) . x0 in W implies ex V being Subset of X0 st
( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W ) )
assume that
A2: W is open and
A3: ((modid (X,A)) | X0) . x0 in W ; ::_thesis: ex V being Subset of X0 st
( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W )
W in A -extension_of_the_topology_of X by A2, PRE_TOPC:def_2;
then consider H, G being Subset of X such that
A4: W = H \/ (G /\ A) and
A5: ( H in the topology of X & G in the topology of X ) ;
reconsider H = H, G = G as Subset of X ;
A6: (H /\ A) \/ (G /\ A) c= W by A4, XBOOLE_1:9, XBOOLE_1:17;
((modid (X,A)) | X0) . x0 = (id the carrier of X) . x by FUNCT_1:49
.= x by FUNCT_1:18 ;
then ( x in H or x in G /\ A ) by A3, A4, XBOOLE_0:def_3;
then ( x in H /\ A or x in G /\ A ) by A1, XBOOLE_0:def_4;
then A7: x in (H /\ A) \/ (G /\ A) by XBOOLE_0:def_3;
A8: ((modid (X,A)) | X0) .: ((H \/ G) /\ A) = (id the carrier of X) .: ((H \/ G) /\ A) by A1, FUNCT_2:97, XBOOLE_1:17
.= (H \/ G) /\ A by FUNCT_1:92 ;
thus ex V being Subset of X0 st
( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W ) ::_thesis: verum
proof
reconsider V = (H \/ G) /\ A as Subset of X0 by A1, XBOOLE_1:17;
take V ; ::_thesis: ( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W )
( H is open & G is open ) by A5, PRE_TOPC:def_2;
then A9: H \/ G is open ;
V = (H \/ G) /\ ([#] X0) by A1;
hence ( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W ) by A7, A8, A6, A9, TOPS_2:24, XBOOLE_1:23; ::_thesis: verum
end;
end;
hence (modid (X,A)) | X0 is_continuous_at x0 by Th43; ::_thesis: verum
end;
theorem Th99: :: TMAP_1:99
for X being non empty TopSpace
for A being Subset of X
for X0 being non empty SubSpace of X st the carrier of X0 misses A holds
(modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A)
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X
for X0 being non empty SubSpace of X st the carrier of X0 misses A holds
(modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A)
let A be Subset of X; ::_thesis: for X0 being non empty SubSpace of X st the carrier of X0 misses A holds
(modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A)
let X0 be non empty SubSpace of X; ::_thesis: ( the carrier of X0 misses A implies (modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A) )
assume the carrier of X0 misses A ; ::_thesis: (modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A)
then for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0 by Th97;
hence (modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A) by Th44; ::_thesis: verum
end;
theorem Th100: :: TMAP_1:100
for X being non empty TopSpace
for A being Subset of X
for X0 being non empty SubSpace of X st the carrier of X0 = A holds
(modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A)
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X
for X0 being non empty SubSpace of X st the carrier of X0 = A holds
(modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A)
let A be Subset of X; ::_thesis: for X0 being non empty SubSpace of X st the carrier of X0 = A holds
(modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A)
let X0 be non empty SubSpace of X; ::_thesis: ( the carrier of X0 = A implies (modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A) )
assume the carrier of X0 = A ; ::_thesis: (modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A)
then for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0 by Th98;
hence (modid (X,A)) | X0 is continuous Function of X0,(X modified_with_respect_to A) by Th44; ::_thesis: verum
end;
theorem Th101: :: TMAP_1:101
for X being non empty TopSpace
for A being Subset of X holds
( A is open iff modid (X,A) is continuous Function of X,(X modified_with_respect_to A) )
proof
let X be non empty TopSpace; ::_thesis: for A being Subset of X holds
( A is open iff modid (X,A) is continuous Function of X,(X modified_with_respect_to A) )
let A be Subset of X; ::_thesis: ( A is open iff modid (X,A) is continuous Function of X,(X modified_with_respect_to A) )
thus ( A is open implies modid (X,A) is continuous Function of X,(X modified_with_respect_to A) ) ::_thesis: ( modid (X,A) is continuous Function of X,(X modified_with_respect_to A) implies A is open )
proof
reconsider f = modid (X,A) as Function of X,X ;
A1: f = id X ;
assume A is open ; ::_thesis: modid (X,A) is continuous Function of X,(X modified_with_respect_to A)
then TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A by Th95;
hence modid (X,A) is continuous Function of X,(X modified_with_respect_to A) by A1, Th51; ::_thesis: verum
end;
A2: [#] (X modified_with_respect_to A) <> {} ;
thus ( modid (X,A) is continuous Function of X,(X modified_with_respect_to A) implies A is open ) ::_thesis: verum
proof
set B = (modid (X,A)) .: A;
assume A3: modid (X,A) is continuous Function of X,(X modified_with_respect_to A) ; ::_thesis: A is open
(modid (X,A)) .: A = A by FUNCT_1:92;
then A4: (modid (X,A)) " ((modid (X,A)) .: A) = A by FUNCT_2:94;
(modid (X,A)) .: A is open by Th94, FUNCT_1:92;
hence A is open by A2, A3, A4, TOPS_2:43; ::_thesis: verum
end;
end;
definition
let X be non empty TopSpace;
let X0 be SubSpace of X;
funcX modified_with_respect_to X0 -> strict TopSpace means :Def10: :: TMAP_1:def 10
for A being Subset of X st A = the carrier of X0 holds
it = X modified_with_respect_to A;
existence
ex b1 being strict TopSpace st
for A being Subset of X st A = the carrier of X0 holds
b1 = X modified_with_respect_to A
proof
reconsider B = the carrier of X0 as Subset of X by TSEP_1:1;
take X modified_with_respect_to B ; ::_thesis: for A being Subset of X st A = the carrier of X0 holds
X modified_with_respect_to B = X modified_with_respect_to A
thus for A being Subset of X st A = the carrier of X0 holds
X modified_with_respect_to B = X modified_with_respect_to A ; ::_thesis: verum
end;
uniqueness
for b1, b2 being strict TopSpace st ( for A being Subset of X st A = the carrier of X0 holds
b1 = X modified_with_respect_to A ) & ( for A being Subset of X st A = the carrier of X0 holds
b2 = X modified_with_respect_to A ) holds
b1 = b2
proof
reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
let Y1, Y2 be strict TopSpace; ::_thesis: ( ( for A being Subset of X st A = the carrier of X0 holds
Y1 = X modified_with_respect_to A ) & ( for A being Subset of X st A = the carrier of X0 holds
Y2 = X modified_with_respect_to A ) implies Y1 = Y2 )
assume that
A1: for A being Subset of X st A = the carrier of X0 holds
Y1 = X modified_with_respect_to A and
A2: for A being Subset of X st A = the carrier of X0 holds
Y2 = X modified_with_respect_to A ; ::_thesis: Y1 = Y2
Y1 = X modified_with_respect_to C by A1;
hence Y1 = Y2 by A2; ::_thesis: verum
end;
end;
:: deftheorem Def10 defines modified_with_respect_to TMAP_1:def_10_:_
for X being non empty TopSpace
for X0 being SubSpace of X
for b3 being strict TopSpace holds
( b3 = X modified_with_respect_to X0 iff for A being Subset of X st A = the carrier of X0 holds
b3 = X modified_with_respect_to A );
registration
let X be non empty TopSpace;
let X0 be SubSpace of X;
clusterX modified_with_respect_to X0 -> non empty strict ;
coherence
not X modified_with_respect_to X0 is empty
proof
[#] X0 c= [#] X by PRE_TOPC:def_4;
then reconsider O = [#] X0 as Subset of X ;
X modified_with_respect_to X0 = X modified_with_respect_to O by Def10;
hence not X modified_with_respect_to X0 is empty ; ::_thesis: verum
end;
end;
theorem Th102: :: TMAP_1:102
for X being non empty TopSpace
for X0 being non empty SubSpace of X holds
( the carrier of (X modified_with_respect_to X0) = the carrier of X & ( for A being Subset of X st A = the carrier of X0 holds
the topology of (X modified_with_respect_to X0) = A -extension_of_the_topology_of X ) )
proof
let X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X holds
( the carrier of (X modified_with_respect_to X0) = the carrier of X & ( for A being Subset of X st A = the carrier of X0 holds
the topology of (X modified_with_respect_to X0) = A -extension_of_the_topology_of X ) )
let X0 be non empty SubSpace of X; ::_thesis: ( the carrier of (X modified_with_respect_to X0) = the carrier of X & ( for A being Subset of X st A = the carrier of X0 holds
the topology of (X modified_with_respect_to X0) = A -extension_of_the_topology_of X ) )
set Y = X modified_with_respect_to X0;
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
A1: X modified_with_respect_to X0 = X modified_with_respect_to A by Def10;
hence the carrier of (X modified_with_respect_to X0) = the carrier of X ; ::_thesis: for A being Subset of X st A = the carrier of X0 holds
the topology of (X modified_with_respect_to X0) = A -extension_of_the_topology_of X
thus for A being Subset of X st A = the carrier of X0 holds
the topology of (X modified_with_respect_to X0) = A -extension_of_the_topology_of X by A1; ::_thesis: verum
end;
theorem :: TMAP_1:103
for X being non empty TopSpace
for X0 being non empty SubSpace of X
for Y0 being SubSpace of X modified_with_respect_to X0 st the carrier of Y0 = the carrier of X0 holds
Y0 is open SubSpace of X modified_with_respect_to X0
proof
let X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for Y0 being SubSpace of X modified_with_respect_to X0 st the carrier of Y0 = the carrier of X0 holds
Y0 is open SubSpace of X modified_with_respect_to X0
let X0 be non empty SubSpace of X; ::_thesis: for Y0 being SubSpace of X modified_with_respect_to X0 st the carrier of Y0 = the carrier of X0 holds
Y0 is open SubSpace of X modified_with_respect_to X0
let Y0 be SubSpace of X modified_with_respect_to X0; ::_thesis: ( the carrier of Y0 = the carrier of X0 implies Y0 is open SubSpace of X modified_with_respect_to X0 )
assume A1: the carrier of Y0 = the carrier of X0 ; ::_thesis: Y0 is open SubSpace of X modified_with_respect_to X0
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
set Y = X modified_with_respect_to X0;
reconsider B = the carrier of Y0 as Subset of (X modified_with_respect_to X0) by TSEP_1:1;
X modified_with_respect_to X0 = X modified_with_respect_to A by Def10;
then B is open by A1, Th94;
hence Y0 is open SubSpace of X modified_with_respect_to X0 by TSEP_1:16; ::_thesis: verum
end;
theorem :: TMAP_1:104
for X being non empty TopSpace
for X0 being non empty SubSpace of X holds
( X0 is open SubSpace of X iff TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to X0 )
proof
let X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X holds
( X0 is open SubSpace of X iff TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to X0 )
let X0 be non empty SubSpace of X; ::_thesis: ( X0 is open SubSpace of X iff TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to X0 )
thus ( X0 is open SubSpace of X implies TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to X0 ) ::_thesis: ( TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to X0 implies X0 is open SubSpace of X )
proof
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
assume X0 is open SubSpace of X ; ::_thesis: TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to X0
then A is open by TSEP_1:def_1;
then TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A by Th95;
hence TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to X0 by Def10; ::_thesis: verum
end;
thus ( TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to X0 implies X0 is open SubSpace of X ) ::_thesis: verum
proof
assume A1: TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to X0 ; ::_thesis: X0 is open SubSpace of X
now__::_thesis:_for_A_being_Subset_of_X_st_A_=_the_carrier_of_X0_holds_
A_is_open
let A be Subset of X; ::_thesis: ( A = the carrier of X0 implies A is open )
assume A = the carrier of X0 ; ::_thesis: A is open
then TopStruct(# the carrier of X, the topology of X #) = X modified_with_respect_to A by A1, Def10;
hence A is open by Th95; ::_thesis: verum
end;
hence X0 is open SubSpace of X by TSEP_1:def_1; ::_thesis: verum
end;
end;
definition
let X be non empty TopSpace;
let X0 be SubSpace of X;
func modid (X,X0) -> Function of X,(X modified_with_respect_to X0) means :Def11: :: TMAP_1:def 11
for A being Subset of X st A = the carrier of X0 holds
it = modid (X,A);
existence
ex b1 being Function of X,(X modified_with_respect_to X0) st
for A being Subset of X st A = the carrier of X0 holds
b1 = modid (X,A)
proof
reconsider B = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider F = modid (X,B) as Function of X,(X modified_with_respect_to X0) by Def10;
take F ; ::_thesis: for A being Subset of X st A = the carrier of X0 holds
F = modid (X,A)
thus for A being Subset of X st A = the carrier of X0 holds
F = modid (X,A) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of X,(X modified_with_respect_to X0) st ( for A being Subset of X st A = the carrier of X0 holds
b1 = modid (X,A) ) & ( for A being Subset of X st A = the carrier of X0 holds
b2 = modid (X,A) ) holds
b1 = b2
proof
reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
let F1, F2 be Function of X,(X modified_with_respect_to X0); ::_thesis: ( ( for A being Subset of X st A = the carrier of X0 holds
F1 = modid (X,A) ) & ( for A being Subset of X st A = the carrier of X0 holds
F2 = modid (X,A) ) implies F1 = F2 )
assume that
A1: for A being Subset of X st A = the carrier of X0 holds
F1 = modid (X,A) and
A2: for A being Subset of X st A = the carrier of X0 holds
F2 = modid (X,A) ; ::_thesis: F1 = F2
F1 = modid (X,C) by A1;
hence F1 = F2 by A2; ::_thesis: verum
end;
end;
:: deftheorem Def11 defines modid TMAP_1:def_11_:_
for X being non empty TopSpace
for X0 being SubSpace of X
for b3 being Function of X,(X modified_with_respect_to X0) holds
( b3 = modid (X,X0) iff for A being Subset of X st A = the carrier of X0 holds
b3 = modid (X,A) );
theorem :: TMAP_1:105
for X being non empty TopSpace
for X0 being non empty SubSpace of X holds modid (X,X0) = id X
proof
let X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X holds modid (X,X0) = id X
let X0 be non empty SubSpace of X; ::_thesis: modid (X,X0) = id X
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
modid (X,A) = modid (X,X0) by Def11;
hence modid (X,X0) = id X ; ::_thesis: verum
end;
theorem Th106: :: TMAP_1:106
for X being non empty TopSpace
for X0, X1 being non empty SubSpace of X st X0 misses X1 holds
for x1 being Point of X1 holds (modid (X,X0)) | X1 is_continuous_at x1
proof
let X be non empty TopSpace; ::_thesis: for X0, X1 being non empty SubSpace of X st X0 misses X1 holds
for x1 being Point of X1 holds (modid (X,X0)) | X1 is_continuous_at x1
let X0, X1 be non empty SubSpace of X; ::_thesis: ( X0 misses X1 implies for x1 being Point of X1 holds (modid (X,X0)) | X1 is_continuous_at x1 )
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider f = (modid (X,A)) | X1 as Function of X1,(X modified_with_respect_to X0) by Def10;
assume A1: X0 misses X1 ; ::_thesis: for x1 being Point of X1 holds (modid (X,X0)) | X1 is_continuous_at x1
let x1 be Point of X1; ::_thesis: (modid (X,X0)) | X1 is_continuous_at x1
the carrier of X1 misses A by A1, TSEP_1:def_3;
then A2: (modid (X,A)) | X1 is_continuous_at x1 by Th97;
now__::_thesis:_for_W_being_Subset_of_(X_modified_with_respect_to_X0)_st_W_is_open_&_f_._x1_in_W_holds_
ex_V_being_Subset_of_X1_st_
(_V_is_open_&_x1_in_V_&_f_.:_V_c=_W_)
let W be Subset of (X modified_with_respect_to X0); ::_thesis: ( W is open & f . x1 in W implies ex V being Subset of X1 st
( V is open & x1 in V & f .: V c= W ) )
reconsider W0 = W as Subset of (X modified_with_respect_to A) by Th102;
assume that
A3: W is open and
A4: f . x1 in W ; ::_thesis: ex V being Subset of X1 st
( V is open & x1 in V & f .: V c= W )
W in the topology of (X modified_with_respect_to X0) by A3, PRE_TOPC:def_2;
then W in A -extension_of_the_topology_of X by Th102;
then A5: W0 is open by PRE_TOPC:def_2;
thus ex V being Subset of X1 st
( V is open & x1 in V & f .: V c= W ) ::_thesis: verum
proof
consider V being Subset of X1 such that
A6: ( V is open & x1 in V & ((modid (X,A)) | X1) .: V c= W0 ) by A2, A4, A5, Th43;
take V ; ::_thesis: ( V is open & x1 in V & f .: V c= W )
thus ( V is open & x1 in V & f .: V c= W ) by A6; ::_thesis: verum
end;
end;
then f is_continuous_at x1 by Th43;
hence (modid (X,X0)) | X1 is_continuous_at x1 by Def11; ::_thesis: verum
end;
theorem Th107: :: TMAP_1:107
for X being non empty TopSpace
for X0 being non empty SubSpace of X
for x0 being Point of X0 holds (modid (X,X0)) | X0 is_continuous_at x0
proof
let X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X
for x0 being Point of X0 holds (modid (X,X0)) | X0 is_continuous_at x0
let X0 be non empty SubSpace of X; ::_thesis: for x0 being Point of X0 holds (modid (X,X0)) | X0 is_continuous_at x0
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider f = (modid (X,A)) | X0 as Function of X0,(X modified_with_respect_to X0) by Def10;
let x0 be Point of X0; ::_thesis: (modid (X,X0)) | X0 is_continuous_at x0
A1: (modid (X,A)) | X0 is_continuous_at x0 by Th98;
now__::_thesis:_for_W_being_Subset_of_(X_modified_with_respect_to_X0)_st_W_is_open_&_f_._x0_in_W_holds_
ex_V_being_Subset_of_X0_st_
(_V_is_open_&_x0_in_V_&_f_.:_V_c=_W_)
let W be Subset of (X modified_with_respect_to X0); ::_thesis: ( W is open & f . x0 in W implies ex V being Subset of X0 st
( V is open & x0 in V & f .: V c= W ) )
reconsider W0 = W as Subset of (X modified_with_respect_to A) by Th102;
assume that
A2: W is open and
A3: f . x0 in W ; ::_thesis: ex V being Subset of X0 st
( V is open & x0 in V & f .: V c= W )
W in the topology of (X modified_with_respect_to X0) by A2, PRE_TOPC:def_2;
then W in A -extension_of_the_topology_of X by Th102;
then A4: W0 is open by PRE_TOPC:def_2;
thus ex V being Subset of X0 st
( V is open & x0 in V & f .: V c= W ) ::_thesis: verum
proof
consider V being Subset of X0 such that
A5: ( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W0 ) by A1, A3, A4, Th43;
take V ; ::_thesis: ( V is open & x0 in V & f .: V c= W )
thus ( V is open & x0 in V & f .: V c= W ) by A5; ::_thesis: verum
end;
end;
then f is_continuous_at x0 by Th43;
hence (modid (X,X0)) | X0 is_continuous_at x0 by Def11; ::_thesis: verum
end;
theorem :: TMAP_1:108
for X being non empty TopSpace
for X0, X1 being non empty SubSpace of X st X0 misses X1 holds
(modid (X,X0)) | X1 is continuous Function of X1,(X modified_with_respect_to X0)
proof
let X be non empty TopSpace; ::_thesis: for X0, X1 being non empty SubSpace of X st X0 misses X1 holds
(modid (X,X0)) | X1 is continuous Function of X1,(X modified_with_respect_to X0)
let X0, X1 be non empty SubSpace of X; ::_thesis: ( X0 misses X1 implies (modid (X,X0)) | X1 is continuous Function of X1,(X modified_with_respect_to X0) )
assume X0 misses X1 ; ::_thesis: (modid (X,X0)) | X1 is continuous Function of X1,(X modified_with_respect_to X0)
then for x1 being Point of X1 holds (modid (X,X0)) | X1 is_continuous_at x1 by Th106;
hence (modid (X,X0)) | X1 is continuous Function of X1,(X modified_with_respect_to X0) by Th44; ::_thesis: verum
end;
theorem :: TMAP_1:109
for X being non empty TopSpace
for X0 being non empty SubSpace of X holds (modid (X,X0)) | X0 is continuous Function of X0,(X modified_with_respect_to X0)
proof
let X be non empty TopSpace; ::_thesis: for X0 being non empty SubSpace of X holds (modid (X,X0)) | X0 is continuous Function of X0,(X modified_with_respect_to X0)
let X0 be non empty SubSpace of X; ::_thesis: (modid (X,X0)) | X0 is continuous Function of X0,(X modified_with_respect_to X0)
for x0 being Point of X0 holds (modid (X,X0)) | X0 is_continuous_at x0 by Th107;
hence (modid (X,X0)) | X0 is continuous Function of X0,(X modified_with_respect_to X0) by Th44; ::_thesis: verum
end;
theorem :: TMAP_1:110
for X being non empty TopSpace
for X0 being SubSpace of X holds
( X0 is open SubSpace of X iff modid (X,X0) is continuous Function of X,(X modified_with_respect_to X0) )
proof
let X be non empty TopSpace; ::_thesis: for X0 being SubSpace of X holds
( X0 is open SubSpace of X iff modid (X,X0) is continuous Function of X,(X modified_with_respect_to X0) )
let X0 be SubSpace of X; ::_thesis: ( X0 is open SubSpace of X iff modid (X,X0) is continuous Function of X,(X modified_with_respect_to X0) )
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
thus ( X0 is open SubSpace of X implies modid (X,X0) is continuous Function of X,(X modified_with_respect_to X0) ) ::_thesis: ( modid (X,X0) is continuous Function of X,(X modified_with_respect_to X0) implies X0 is open SubSpace of X )
proof
assume X0 is open SubSpace of X ; ::_thesis: modid (X,X0) is continuous Function of X,(X modified_with_respect_to X0)
then A1: A is open by TSEP_1:16;
( X modified_with_respect_to X0 = X modified_with_respect_to A & modid (X,X0) = modid (X,A) ) by Def10, Def11;
hence modid (X,X0) is continuous Function of X,(X modified_with_respect_to X0) by A1, Th101; ::_thesis: verum
end;
thus ( modid (X,X0) is continuous Function of X,(X modified_with_respect_to X0) implies X0 is open SubSpace of X ) ::_thesis: verum
proof
assume A2: modid (X,X0) is continuous Function of X,(X modified_with_respect_to X0) ; ::_thesis: X0 is open SubSpace of X
( X modified_with_respect_to X0 = X modified_with_respect_to A & modid (X,X0) = modid (X,A) ) by Def10, Def11;
then A is open by A2, Th101;
hence X0 is open SubSpace of X by TSEP_1:16; ::_thesis: verum
end;
end;
begin
theorem Th111: :: TMAP_1:111
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for g being Function of (X1 union X2),Y
for x1 being Point of X1
for x2 being Point of X2
for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for g being Function of (X1 union X2),Y
for x1 being Point of X1
for x2 being Point of X2
for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: for g being Function of (X1 union X2),Y
for x1 being Point of X1
for x2 being Point of X2
for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
let g be Function of (X1 union X2),Y; ::_thesis: for x1 being Point of X1
for x2 being Point of X2
for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
let x1 be Point of X1; ::_thesis: for x2 being Point of X2
for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
let x2 be Point of X2; ::_thesis: for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
let x be Point of (X1 union X2); ::_thesis: ( x = x1 & x = x2 implies ( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) ) )
assume that
A1: x = x1 and
A2: x = x2 ; ::_thesis: ( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
A3: X2 is SubSpace of X1 union X2 by TSEP_1:22;
A4: X1 is SubSpace of X1 union X2 by TSEP_1:22;
hence ( g is_continuous_at x implies ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) ) by A1, A2, A3, Th74; ::_thesis: ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 implies g is_continuous_at x )
thus ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 implies g is_continuous_at x ) ::_thesis: verum
proof
assume that
A5: g | X1 is_continuous_at x1 and
A6: g | X2 is_continuous_at x2 ; ::_thesis: g is_continuous_at x
for G being a_neighborhood of g . x ex H being a_neighborhood of x st g .: H c= G
proof
let G be a_neighborhood of g . x; ::_thesis: ex H being a_neighborhood of x st g .: H c= G
g . x = (g | X1) . x1 by A1, A4, Th65;
then consider H1 being a_neighborhood of x1 such that
A7: (g | X1) .: H1 c= G by A5, Def2;
g . x = (g | X2) . x2 by A2, A3, Th65;
then consider H2 being a_neighborhood of x2 such that
A8: (g | X2) .: H2 c= G by A6, Def2;
the carrier of X2 c= the carrier of (X1 union X2) by A3, TSEP_1:4;
then reconsider S2 = H2 as Subset of (X1 union X2) by XBOOLE_1:1;
g .: S2 c= G by A3, A8, Th68;
then A9: S2 c= g " G by FUNCT_2:95;
the carrier of X1 c= the carrier of (X1 union X2) by A4, TSEP_1:4;
then reconsider S1 = H1 as Subset of (X1 union X2) by XBOOLE_1:1;
consider H being a_neighborhood of x such that
A10: H c= H1 \/ H2 by A1, A2, Th16;
take H ; ::_thesis: g .: H c= G
g .: S1 c= G by A4, A7, Th68;
then S1 c= g " G by FUNCT_2:95;
then S1 \/ S2 c= g " G by A9, XBOOLE_1:8;
then H c= g " G by A10, XBOOLE_1:1;
hence g .: H c= G by FUNCT_2:95; ::_thesis: verum
end;
hence g is_continuous_at x by Def2; ::_thesis: verum
end;
end;
theorem :: TMAP_1:112
for X, Y being non empty TopSpace
for f being Function of X,Y
for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
let f be Function of X,Y; ::_thesis: for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: for x being Point of (X1 union X2)
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
A1: ( X1 is SubSpace of X1 union X2 & X2 is SubSpace of X1 union X2 ) by TSEP_1:22;
let x be Point of (X1 union X2); ::_thesis: for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
let x1 be Point of X1; ::_thesis: for x2 being Point of X2 st x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
let x2 be Point of X2; ::_thesis: ( x = x1 & x = x2 implies ( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) )
assume A2: ( x = x1 & x = x2 ) ; ::_thesis: ( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
thus ( f | (X1 union X2) is_continuous_at x implies ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) by A2, A1, Th75; ::_thesis: ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f | (X1 union X2) is_continuous_at x )
thus ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f | (X1 union X2) is_continuous_at x ) ::_thesis: verum
proof
set g = f | (X1 union X2);
assume A3: ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ; ::_thesis: f | (X1 union X2) is_continuous_at x
( (f | (X1 union X2)) | X1 = f | X1 & (f | (X1 union X2)) | X2 = f | X2 ) by A1, Th70;
hence f | (X1 union X2) is_continuous_at x by A2, A3, Th111; ::_thesis: verum
end;
end;
theorem :: TMAP_1:113
for X, Y being non empty TopSpace
for f being Function of X,Y
for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
for x being Point of X
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
for x being Point of X
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
let f be Function of X,Y; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
for x being Point of X
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 implies for x being Point of X
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) )
assume A1: X = X1 union X2 ; ::_thesis: for x being Point of X
for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
let x be Point of X; ::_thesis: for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
let x1 be Point of X1; ::_thesis: for x2 being Point of X2 st x = x1 & x = x2 holds
( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
let x2 be Point of X2; ::_thesis: ( x = x1 & x = x2 implies ( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) )
assume that
A2: x = x1 and
A3: x = x2 ; ::_thesis: ( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
thus ( f is_continuous_at x implies ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) by A2, A3, Th58; ::_thesis: ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f is_continuous_at x )
thus ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f is_continuous_at x ) ::_thesis: verum
proof
assume that
A4: f | X1 is_continuous_at x1 and
A5: f | X2 is_continuous_at x2 ; ::_thesis: f is_continuous_at x
for G being a_neighborhood of f . x ex H being a_neighborhood of x st f .: H c= G
proof
let G be a_neighborhood of f . x; ::_thesis: ex H being a_neighborhood of x st f .: H c= G
f . x = (f | X1) . x1 by A2, FUNCT_1:49;
then consider H1 being a_neighborhood of x1 such that
A6: (f | X1) .: H1 c= G by A4, Def2;
the carrier of X1 c= the carrier of X by BORSUK_1:1;
then reconsider S1 = H1 as Subset of X by XBOOLE_1:1;
f . x = (f | X2) . x2 by A3, FUNCT_1:49;
then consider H2 being a_neighborhood of x2 such that
A7: (f | X2) .: H2 c= G by A5, Def2;
the carrier of X2 c= the carrier of X by BORSUK_1:1;
then reconsider S2 = H2 as Subset of X by XBOOLE_1:1;
f .: S2 c= G by A7, FUNCT_2:97;
then A8: S2 c= f " G by FUNCT_2:95;
consider H being a_neighborhood of x such that
A9: H c= H1 \/ H2 by A1, A2, A3, Th16;
take H ; ::_thesis: f .: H c= G
f .: S1 c= G by A6, FUNCT_2:97;
then S1 c= f " G by FUNCT_2:95;
then S1 \/ S2 c= f " G by A8, XBOOLE_1:8;
then H c= f " G by A9, XBOOLE_1:1;
hence f .: H c= G by FUNCT_2:95; ::_thesis: verum
end;
hence f is_continuous_at x by Def2; ::_thesis: verum
end;
end;
theorem Th114: :: TMAP_1:114
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1,X2 are_weakly_separated holds
for g being Function of (X1 union X2),Y holds
( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1,X2 are_weakly_separated holds
for g being Function of (X1 union X2),Y holds
( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_weakly_separated implies for g being Function of (X1 union X2),Y holds
( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) ) )
assume A1: X1,X2 are_weakly_separated ; ::_thesis: for g being Function of (X1 union X2),Y holds
( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
let g be Function of (X1 union X2),Y; ::_thesis: ( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
A2: X2 is SubSpace of X1 union X2 by TSEP_1:22;
A3: X1 is SubSpace of X1 union X2 by TSEP_1:22;
hence ( g is continuous Function of (X1 union X2),Y implies ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) ) by A2, Th82; ::_thesis: ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y implies g is continuous Function of (X1 union X2),Y )
thus ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y implies g is continuous Function of (X1 union X2),Y ) ::_thesis: verum
proof
assume that
A4: g | X1 is continuous Function of X1,Y and
A5: g | X2 is continuous Function of X2,Y ; ::_thesis: g is continuous Function of (X1 union X2),Y
for x being Point of (X1 union X2) holds g is_continuous_at x
proof
set X0 = X1 union X2;
let x be Point of (X1 union X2); ::_thesis: g is_continuous_at x
A6: ( X1 meets X2 implies g is_continuous_at x )
proof
assume A7: X1 meets X2 ; ::_thesis: g is_continuous_at x
A8: now__::_thesis:_(_X1_is_not_SubSpace_of_X2_&_X2_is_not_SubSpace_of_X1_implies_g_is_continuous_at_x_)
assume A9: ( X1 is not SubSpace of X2 & X2 is not SubSpace of X1 ) ; ::_thesis: g is_continuous_at x
then consider Y1, Y2 being non empty open SubSpace of X such that
A10: Y1 meet (X1 union X2) is SubSpace of X1 and
A11: Y2 meet (X1 union X2) is SubSpace of X2 and
A12: ( X1 union X2 is SubSpace of Y1 union Y2 or ex Z being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Z & Z meet (X1 union X2) is SubSpace of X1 meet X2 ) ) by A1, A7, TSEP_1:89;
A13: ( Y2 meets X1 union X2 implies Y2 meet (X1 union X2) is open SubSpace of X1 union X2 ) by Th39;
A14: ( Y1 meets X1 union X2 implies Y1 meet (X1 union X2) is open SubSpace of X1 union X2 ) by Th39;
A15: now__::_thesis:_(_X1_union_X2_is_not_SubSpace_of_Y1_union_Y2_implies_g_is_continuous_at_x_)
X is SubSpace of X by TSEP_1:2;
then reconsider X12 = TopStruct(# the carrier of X, the topology of X #) as SubSpace of X by Th6;
assume A16: X1 union X2 is not SubSpace of Y1 union Y2 ; ::_thesis: g is_continuous_at x
then consider Z being non empty closed SubSpace of X such that
A17: TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Z and
A18: Z meet (X1 union X2) is SubSpace of X1 meet X2 by A12;
the carrier of (X1 union X2) c= the carrier of X12 by BORSUK_1:1;
then A19: X1 union X2 is SubSpace of X12 by TSEP_1:4;
then X12 meets X1 union X2 by Th17;
then A20: ((Y1 union Y2) union Z) meet (X1 union X2) = TopStruct(# the carrier of (X1 union X2), the topology of (X1 union X2) #) by A17, A19, TSEP_1:28;
A21: ( Y1 meets X1 union X2 & Y2 meets X1 union X2 ) by A7, A9, A10, A11, A17, A18, Th32;
A22: now__::_thesis:_(_ex_x12_being_Point_of_((Y1_union_Y2)_meet_(X1_union_X2))_st_x12_=_x_implies_g_is_continuous_at_x_)
A23: now__::_thesis:_(_ex_x2_being_Point_of_(Y2_meet_(X1_union_X2))_st_x2_=_x_implies_g_is_continuous_at_x_)
given x2 being Point of (Y2 meet (X1 union X2)) such that A24: x2 = x ; ::_thesis: g is_continuous_at x
g | (Y2 meet (X1 union X2)) is continuous by A2, A5, A11, Th83;
then g | (Y2 meet (X1 union X2)) is_continuous_at x2 by Th44;
hence g is_continuous_at x by A7, A9, A10, A11, A13, A17, A18, A24, Th32, Th79; ::_thesis: verum
end;
A25: now__::_thesis:_(_ex_x1_being_Point_of_(Y1_meet_(X1_union_X2))_st_x1_=_x_implies_g_is_continuous_at_x_)
given x1 being Point of (Y1 meet (X1 union X2)) such that A26: x1 = x ; ::_thesis: g is_continuous_at x
g | (Y1 meet (X1 union X2)) is continuous by A3, A4, A10, Th83;
then g | (Y1 meet (X1 union X2)) is_continuous_at x1 by Th44;
hence g is_continuous_at x by A7, A9, A10, A11, A14, A17, A18, A26, Th32, Th79; ::_thesis: verum
end;
assume A27: ex x12 being Point of ((Y1 union Y2) meet (X1 union X2)) st x12 = x ; ::_thesis: g is_continuous_at x
(Y1 union Y2) meet (X1 union X2) = (Y1 meet (X1 union X2)) union (Y2 meet (X1 union X2)) by A21, TSEP_1:32;
hence g is_continuous_at x by A27, A25, A23, Th11; ::_thesis: verum
end;
A28: now__::_thesis:_(_ex_x0_being_Point_of_(Z_meet_(X1_union_X2))_st_x0_=_x_implies_g_is_continuous_at_x_)
given x0 being Point of (Z meet (X1 union X2)) such that A29: x0 = x ; ::_thesis: g is_continuous_at x
consider x00 being Point of (X1 meet X2) such that
A30: x00 = x0 by A18, Th10;
consider x1 being Point of X1 such that
A31: x1 = x00 by A7, Th12;
consider x2 being Point of X2 such that
A32: x2 = x00 by A7, Th12;
( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) by A4, A5, Th44;
hence g is_continuous_at x by A29, A30, A31, A32, Th111; ::_thesis: verum
end;
( Y1 union Y2 meets X1 union X2 & Z meets X1 union X2 ) by A7, A9, A10, A11, A16, A17, A18, Th33;
then ((Y1 union Y2) meet (X1 union X2)) union (Z meet (X1 union X2)) = TopStruct(# the carrier of (X1 union X2), the topology of (X1 union X2) #) by A20, TSEP_1:32;
hence g is_continuous_at x by A22, A28, Th11; ::_thesis: verum
end;
now__::_thesis:_(_X1_union_X2_is_SubSpace_of_Y1_union_Y2_implies_g_is_continuous_at_x_)
assume A33: X1 union X2 is SubSpace of Y1 union Y2 ; ::_thesis: g is_continuous_at x
then A34: Y1 meets X1 union X2 by A9, A10, A11, Th31;
A35: now__::_thesis:_(_ex_x2_being_Point_of_(Y2_meet_(X1_union_X2))_st_x2_=_x_implies_g_is_continuous_at_x_)
given x2 being Point of (Y2 meet (X1 union X2)) such that A36: x2 = x ; ::_thesis: g is_continuous_at x
g | (Y2 meet (X1 union X2)) is continuous by A2, A5, A11, Th83;
then g | (Y2 meet (X1 union X2)) is_continuous_at x2 by Th44;
hence g is_continuous_at x by A9, A10, A11, A13, A33, A36, Th31, Th79; ::_thesis: verum
end;
A37: now__::_thesis:_(_ex_x1_being_Point_of_(Y1_meet_(X1_union_X2))_st_x1_=_x_implies_g_is_continuous_at_x_)
given x1 being Point of (Y1 meet (X1 union X2)) such that A38: x1 = x ; ::_thesis: g is_continuous_at x
g | (Y1 meet (X1 union X2)) is continuous by A3, A4, A10, Th83;
then g | (Y1 meet (X1 union X2)) is_continuous_at x1 by Th44;
hence g is_continuous_at x by A9, A10, A11, A14, A33, A38, Th31, Th79; ::_thesis: verum
end;
Y1 is SubSpace of Y1 union Y2 by TSEP_1:22;
then Y1 union Y2 meets X1 union X2 by A34, Th18;
then A39: (Y1 union Y2) meet (X1 union X2) = X1 union X2 by A33, TSEP_1:28;
Y2 meets X1 union X2 by A9, A10, A11, A33, Th31;
then (Y1 meet (X1 union X2)) union (Y2 meet (X1 union X2)) = X1 union X2 by A34, A39, TSEP_1:32;
hence g is_continuous_at x by A37, A35, Th11; ::_thesis: verum
end;
hence g is_continuous_at x by A15; ::_thesis: verum
end;
now__::_thesis:_(_(_X1_is_SubSpace_of_X2_or_X2_is_SubSpace_of_X1_)_implies_g_is_continuous_at_x_)
A40: now__::_thesis:_(_X2_is_SubSpace_of_X1_implies_g_is_continuous_at_x_)
assume X2 is SubSpace of X1 ; ::_thesis: g is_continuous_at x
then A41: TopStruct(# the carrier of X1, the topology of X1 #) = X1 union X2 by TSEP_1:23;
then reconsider x1 = x as Point of X1 ;
g | X1 is_continuous_at x1 by A4, Th44;
hence g is_continuous_at x by A41, Th81; ::_thesis: verum
end;
A42: now__::_thesis:_(_X1_is_SubSpace_of_X2_implies_g_is_continuous_at_x_)
assume X1 is SubSpace of X2 ; ::_thesis: g is_continuous_at x
then A43: TopStruct(# the carrier of X2, the topology of X2 #) = X1 union X2 by TSEP_1:23;
then reconsider x2 = x as Point of X2 ;
g | X2 is_continuous_at x2 by A5, Th44;
hence g is_continuous_at x by A43, Th81; ::_thesis: verum
end;
assume ( X1 is SubSpace of X2 or X2 is SubSpace of X1 ) ; ::_thesis: g is_continuous_at x
hence g is_continuous_at x by A42, A40; ::_thesis: verum
end;
hence g is_continuous_at x by A8; ::_thesis: verum
end;
( X1 misses X2 implies g is_continuous_at x )
proof
assume X1 misses X2 ; ::_thesis: g is_continuous_at x
then X1,X2 are_separated by A1, TSEP_1:78;
then consider Y1, Y2 being non empty open SubSpace of X such that
A44: X1 is SubSpace of Y1 and
A45: X2 is SubSpace of Y2 and
A46: ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) by TSEP_1:77;
Y2 misses X1 by A44, A45, A46, Th30;
then A47: X2 is open SubSpace of X1 union X2 by A45, Th41;
A48: now__::_thesis:_(_ex_x2_being_Point_of_X2_st_x2_=_x_implies_g_is_continuous_at_x_)
given x2 being Point of X2 such that A49: x2 = x ; ::_thesis: g is_continuous_at x
g | X2 is_continuous_at x2 by A5, Th44;
hence g is_continuous_at x by A47, A49, Th79; ::_thesis: verum
end;
Y1 misses X2 by A44, A45, A46, Th30;
then A50: X1 is open SubSpace of X1 union X2 by A44, Th41;
now__::_thesis:_(_ex_x1_being_Point_of_X1_st_x1_=_x_implies_g_is_continuous_at_x_)
given x1 being Point of X1 such that A51: x1 = x ; ::_thesis: g is_continuous_at x
g | X1 is_continuous_at x1 by A4, Th44;
hence g is_continuous_at x by A50, A51, Th79; ::_thesis: verum
end;
hence g is_continuous_at x by A48, Th11; ::_thesis: verum
end;
hence g is_continuous_at x by A6; ::_thesis: verum
end;
hence g is continuous Function of (X1 union X2),Y by Th44; ::_thesis: verum
end;
end;
theorem Th115: :: TMAP_1:115
for X, Y being non empty TopSpace
for X1, X2 being non empty closed SubSpace of X
for g being Function of (X1 union X2),Y holds
( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty closed SubSpace of X
for g being Function of (X1 union X2),Y holds
( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
let X1, X2 be non empty closed SubSpace of X; ::_thesis: for g being Function of (X1 union X2),Y holds
( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
let g be Function of (X1 union X2),Y; ::_thesis: ( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
X1,X2 are_weakly_separated by TSEP_1:80;
hence ( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) ) by Th114; ::_thesis: verum
end;
theorem Th116: :: TMAP_1:116
for X, Y being non empty TopSpace
for X1, X2 being non empty open SubSpace of X
for g being Function of (X1 union X2),Y holds
( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty open SubSpace of X
for g being Function of (X1 union X2),Y holds
( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
let X1, X2 be non empty open SubSpace of X; ::_thesis: for g being Function of (X1 union X2),Y holds
( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
let g be Function of (X1 union X2),Y; ::_thesis: ( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) )
X1,X2 are_weakly_separated by TSEP_1:81;
hence ( g is continuous Function of (X1 union X2),Y iff ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y ) ) by Th114; ::_thesis: verum
end;
theorem Th117: :: TMAP_1:117
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1,X2 are_weakly_separated holds
for f being Function of X,Y holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1,X2 are_weakly_separated holds
for f being Function of X,Y holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_weakly_separated implies for f being Function of X,Y holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) )
assume A1: X1,X2 are_weakly_separated ; ::_thesis: for f being Function of X,Y holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let f be Function of X,Y; ::_thesis: ( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
A2: X2 is SubSpace of X1 union X2 by TSEP_1:22;
then A3: (f | (X1 union X2)) | X2 = f | X2 by Th71;
A4: X1 is SubSpace of X1 union X2 by TSEP_1:22;
then A5: (f | (X1 union X2)) | X1 = f | X1 by Th71;
hence ( f | (X1 union X2) is continuous Function of (X1 union X2),Y implies ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) by A4, A2, A3, Th82; ::_thesis: ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y implies f | (X1 union X2) is continuous Function of (X1 union X2),Y )
thus ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y implies f | (X1 union X2) is continuous Function of (X1 union X2),Y ) by A1, A5, A3, Th114; ::_thesis: verum
end;
theorem :: TMAP_1:118
for X, Y being non empty TopSpace
for f being Function of X,Y
for X1, X2 being non empty closed SubSpace of X holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X1, X2 being non empty closed SubSpace of X holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let f be Function of X,Y; ::_thesis: for X1, X2 being non empty closed SubSpace of X holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let X1, X2 be non empty closed SubSpace of X; ::_thesis: ( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
X1,X2 are_weakly_separated by TSEP_1:80;
hence ( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) by Th117; ::_thesis: verum
end;
theorem :: TMAP_1:119
for X, Y being non empty TopSpace
for f being Function of X,Y
for X1, X2 being non empty open SubSpace of X holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X1, X2 being non empty open SubSpace of X holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let f be Function of X,Y; ::_thesis: for X1, X2 being non empty open SubSpace of X holds
( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let X1, X2 be non empty open SubSpace of X; ::_thesis: ( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
X1,X2 are_weakly_separated by TSEP_1:81;
hence ( f | (X1 union X2) is continuous Function of (X1 union X2),Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) by Th117; ::_thesis: verum
end;
theorem Th120: :: TMAP_1:120
for X, Y being non empty TopSpace
for f being Function of X,Y
for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1,X2 are_weakly_separated holds
( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1,X2 are_weakly_separated holds
( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let f be Function of X,Y; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1,X2 are_weakly_separated holds
( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 & X1,X2 are_weakly_separated implies ( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) )
assume that
A1: X = X1 union X2 and
A2: X1,X2 are_weakly_separated ; ::_thesis: ( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
thus ( f is continuous Function of X,Y implies ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) ; ::_thesis: ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y implies f is continuous Function of X,Y )
assume ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ; ::_thesis: f is continuous Function of X,Y
then f | (X1 union X2) is continuous Function of (X1 union X2),Y by A2, Th117;
hence f is continuous Function of X,Y by A1, Th54; ::_thesis: verum
end;
theorem Th121: :: TMAP_1:121
for X, Y being non empty TopSpace
for f being Function of X,Y
for X1, X2 being non empty closed SubSpace of X st X = X1 union X2 holds
( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X1, X2 being non empty closed SubSpace of X st X = X1 union X2 holds
( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let f be Function of X,Y; ::_thesis: for X1, X2 being non empty closed SubSpace of X st X = X1 union X2 holds
( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let X1, X2 be non empty closed SubSpace of X; ::_thesis: ( X = X1 union X2 implies ( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) )
assume A1: X = X1 union X2 ; ::_thesis: ( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
X1,X2 are_weakly_separated by TSEP_1:80;
hence ( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) by A1, Th120; ::_thesis: verum
end;
theorem Th122: :: TMAP_1:122
for X, Y being non empty TopSpace
for f being Function of X,Y
for X1, X2 being non empty open SubSpace of X st X = X1 union X2 holds
( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y
for X1, X2 being non empty open SubSpace of X st X = X1 union X2 holds
( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let f be Function of X,Y; ::_thesis: for X1, X2 being non empty open SubSpace of X st X = X1 union X2 holds
( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
let X1, X2 be non empty open SubSpace of X; ::_thesis: ( X = X1 union X2 implies ( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) )
assume A1: X = X1 union X2 ; ::_thesis: ( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) )
X1,X2 are_weakly_separated by TSEP_1:81;
hence ( f is continuous Function of X,Y iff ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ) by A1, Th120; ::_thesis: verum
end;
theorem Th123: :: TMAP_1:123
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y ) ) )
thus ( X1,X2 are_separated implies ( X1 misses X2 & ( for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y ) ) ) ::_thesis: ( X1 misses X2 & ( for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y ) implies X1,X2 are_separated )
proof
assume A1: X1,X2 are_separated ; ::_thesis: ( X1 misses X2 & ( for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y ) )
hence X1 misses X2 by TSEP_1:63; ::_thesis: for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y
X1,X2 are_weakly_separated by A1, TSEP_1:78;
hence for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y by Th114; ::_thesis: verum
end;
thus ( X1 misses X2 & ( for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y ) implies X1,X2 are_separated ) ::_thesis: verum
proof
reconsider Y1 = X1, Y2 = X2 as SubSpace of X1 union X2 by TSEP_1:22;
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
A2: the carrier of (X1 union X2) = A1 \/ A2 by TSEP_1:def_2;
then reconsider C1 = A1 as Subset of (X1 union X2) by XBOOLE_1:7;
reconsider C2 = A2 as Subset of (X1 union X2) by A2, XBOOLE_1:7;
A3: Cl C1 = (Cl A1) /\ ([#] (X1 union X2)) by PRE_TOPC:17;
A4: Cl C2 = (Cl A2) /\ ([#] (X1 union X2)) by PRE_TOPC:17;
assume X1 misses X2 ; ::_thesis: ( ex Y being non empty TopSpace ex g being Function of (X1 union X2),Y st
( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y & g is not continuous Function of (X1 union X2),Y ) or X1,X2 are_separated )
then A5: C1 misses C2 by TSEP_1:def_3;
assume A6: for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y ; ::_thesis: X1,X2 are_separated
assume X1,X2 are_not_separated ; ::_thesis: contradiction
then A7: ex A10, A20 being Subset of X st
( A10 = the carrier of X1 & A20 = the carrier of X2 & A10,A20 are_not_separated ) by TSEP_1:def_6;
A8: now__::_thesis:_not_C1,C2_are_separated
assume A9: C1,C2 are_separated ; ::_thesis: contradiction
then (Cl A1) /\ ([#] (X1 union X2)) misses A2 by A3, CONNSP_1:def_1;
then ((Cl A1) /\ ([#] (X1 union X2))) /\ A2 = {} by XBOOLE_0:def_7;
then A10: ((Cl A1) /\ A2) /\ ([#] (X1 union X2)) = {} by XBOOLE_1:16;
A1 misses (Cl A2) /\ ([#] (X1 union X2)) by A4, A9, CONNSP_1:def_1;
then A1 /\ ((Cl A2) /\ ([#] (X1 union X2))) = {} by XBOOLE_0:def_7;
then A11: (A1 /\ (Cl A2)) /\ ([#] (X1 union X2)) = {} by XBOOLE_1:16;
( C1 c= [#] (X1 union X2) & A1 /\ (Cl A2) c= A1 ) by XBOOLE_1:17;
then A1 /\ (Cl A2) = {} by A11, XBOOLE_1:1, XBOOLE_1:28;
then A12: A1 misses Cl A2 by XBOOLE_0:def_7;
( C2 c= [#] (X1 union X2) & (Cl A1) /\ A2 c= A2 ) by XBOOLE_1:17;
then (Cl A1) /\ A2 = {} by A10, XBOOLE_1:1, XBOOLE_1:28;
then Cl A1 misses A2 by XBOOLE_0:def_7;
hence contradiction by A7, A12, CONNSP_1:def_1; ::_thesis: verum
end;
now__::_thesis:_contradiction
percases ( not C1 is open or not C2 is open ) by A8, A5, TSEP_1:37;
supposeA13: not C1 is open ; ::_thesis: contradiction
set g = modid ((X1 union X2),C1);
set Y = (X1 union X2) modified_with_respect_to C1;
(modid ((X1 union X2),C1)) | Y1 = (modid ((X1 union X2),C1)) | X1 by Def5;
then A14: (modid ((X1 union X2),C1)) | X1 is continuous Function of X1,((X1 union X2) modified_with_respect_to C1) by Th100;
(modid ((X1 union X2),C1)) | Y2 = (modid ((X1 union X2),C1)) | X2 by Def5;
then A15: (modid ((X1 union X2),C1)) | X2 is continuous Function of X2,((X1 union X2) modified_with_respect_to C1) by A5, Th99;
modid ((X1 union X2),C1) is not continuous Function of (X1 union X2),((X1 union X2) modified_with_respect_to C1) by A13, Th101;
hence contradiction by A6, A14, A15; ::_thesis: verum
end;
supposeA16: not C2 is open ; ::_thesis: contradiction
set g = modid ((X1 union X2),C2);
set Y = (X1 union X2) modified_with_respect_to C2;
(modid ((X1 union X2),C2)) | Y2 = (modid ((X1 union X2),C2)) | X2 by Def5;
then A17: (modid ((X1 union X2),C2)) | X2 is continuous Function of X2,((X1 union X2) modified_with_respect_to C2) by Th100;
(modid ((X1 union X2),C2)) | Y1 = (modid ((X1 union X2),C2)) | X1 by Def5;
then A18: (modid ((X1 union X2),C2)) | X1 is continuous Function of X1,((X1 union X2) modified_with_respect_to C2) by A5, Th99;
modid ((X1 union X2),C2) is not continuous Function of (X1 union X2),((X1 union X2) modified_with_respect_to C2) by A16, Th101;
hence contradiction by A6, A18, A17; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
theorem Th124: :: TMAP_1:124
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y ) ) )
thus ( X1,X2 are_separated implies ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y ) ) ) ::_thesis: ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y ) implies X1,X2 are_separated )
proof
assume A1: X1,X2 are_separated ; ::_thesis: ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y ) )
hence X1 misses X2 by TSEP_1:63; ::_thesis: for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y
X1,X2 are_weakly_separated by A1, TSEP_1:78;
hence for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y by Th117; ::_thesis: verum
end;
thus ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y ) implies X1,X2 are_separated ) ::_thesis: verum
proof
assume A2: X1 misses X2 ; ::_thesis: ( ex Y being non empty TopSpace ex f being Function of X,Y st
( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y & f | (X1 union X2) is not continuous Function of (X1 union X2),Y ) or X1,X2 are_separated )
assume A3: for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y ; ::_thesis: X1,X2 are_separated
for Y being non empty TopSpace
for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y
proof
let Y be non empty TopSpace; ::_thesis: for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y
let g be Function of (X1 union X2),Y; ::_thesis: ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y implies g is continuous Function of (X1 union X2),Y )
assume that
A4: g | X1 is continuous Function of X1,Y and
A5: g | X2 is continuous Function of X2,Y ; ::_thesis: g is continuous Function of (X1 union X2),Y
consider h being Function of X,Y such that
A6: h | (X1 union X2) = g by Th57;
X2 is SubSpace of X1 union X2 by TSEP_1:22;
then A7: h | X2 is continuous Function of X2,Y by A5, A6, Th70;
X1 is SubSpace of X1 union X2 by TSEP_1:22;
then h | X1 is continuous Function of X1,Y by A4, A6, Th70;
hence g is continuous Function of (X1 union X2),Y by A3, A6, A7; ::_thesis: verum
end;
hence X1,X2 are_separated by A2, Th123; ::_thesis: verum
end;
end;
theorem :: TMAP_1:125
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 implies ( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y ) ) ) )
assume A1: X = X1 union X2 ; ::_thesis: ( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y ) ) )
thus ( X1,X2 are_separated implies ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y ) ) ) ::_thesis: ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y ) implies X1,X2 are_separated )
proof
assume A2: X1,X2 are_separated ; ::_thesis: ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y ) )
hence X1 misses X2 by TSEP_1:63; ::_thesis: for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y
X1,X2 are_weakly_separated by A2, TSEP_1:78;
hence for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y by A1, Th120; ::_thesis: verum
end;
thus ( X1 misses X2 & ( for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y ) implies X1,X2 are_separated ) ::_thesis: verum
proof
assume A3: X1 misses X2 ; ::_thesis: ( ex Y being non empty TopSpace ex f being Function of X,Y st
( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y & f is not continuous Function of X,Y ) or X1,X2 are_separated )
assume A4: for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f is continuous Function of X,Y ; ::_thesis: X1,X2 are_separated
for Y being non empty TopSpace
for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y
proof
let Y be non empty TopSpace; ::_thesis: for f being Function of X,Y st f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y holds
f | (X1 union X2) is continuous Function of (X1 union X2),Y
let f be Function of X,Y; ::_thesis: ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y implies f | (X1 union X2) is continuous Function of (X1 union X2),Y )
assume ( f | X1 is continuous Function of X1,Y & f | X2 is continuous Function of X2,Y ) ; ::_thesis: f | (X1 union X2) is continuous Function of (X1 union X2),Y
then f is continuous Function of X,Y by A4;
hence f | (X1 union X2) is continuous Function of (X1 union X2),Y ; ::_thesis: verum
end;
hence X1,X2 are_separated by A3, Th124; ::_thesis: verum
end;
end;
begin
definition
let X, Y be non empty TopSpace;
let X1, X2 be non empty SubSpace of X;
let f1 be Function of X1,Y;
let f2 be Function of X2,Y;
assume A1: ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) ;
funcf1 union f2 -> Function of (X1 union X2),Y means :Def12: :: TMAP_1:def 12
( it | X1 = f1 & it | X2 = f2 );
existence
ex b1 being Function of (X1 union X2),Y st
( b1 | X1 = f1 & b1 | X2 = f2 )
proof
set B = the carrier of Y;
set A = the carrier of (X1 union X2);
set A2 = the carrier of X2;
set A1 = the carrier of X1;
A2: ( X1 is SubSpace of X1 union X2 & X2 is SubSpace of X1 union X2 ) by TSEP_1:22;
A3: the carrier of (X1 union X2) = the carrier of X1 \/ the carrier of X2 by TSEP_1:def_2;
A4: ( the carrier of X1 meets the carrier of X2 implies f1 | ( the carrier of X1 /\ the carrier of X2) = f2 | ( the carrier of X1 /\ the carrier of X2) )
proof
assume A5: the carrier of X1 meets the carrier of X2 ; ::_thesis: f1 | ( the carrier of X1 /\ the carrier of X2) = f2 | ( the carrier of X1 /\ the carrier of X2)
then A6: X1 meets X2 by TSEP_1:def_3;
then A7: X1 meet X2 is SubSpace of X1 by TSEP_1:27;
A8: X1 meet X2 is SubSpace of X2 by A6, TSEP_1:27;
thus f1 | ( the carrier of X1 /\ the carrier of X2) = f1 | the carrier of (X1 meet X2) by A6, TSEP_1:def_4
.= f2 | (X1 meet X2) by A1, A5, A7, Def5, TSEP_1:def_3
.= f2 | the carrier of (X1 meet X2) by A8, Def5
.= f2 | ( the carrier of X1 /\ the carrier of X2) by A6, TSEP_1:def_4 ; ::_thesis: verum
end;
reconsider A1 = the carrier of X1, A2 = the carrier of X2 as non empty Subset of the carrier of (X1 union X2) by A3, XBOOLE_1:7;
reconsider g1 = f1 as Function of A1, the carrier of Y ;
reconsider g2 = f2 as Function of A2, the carrier of Y ;
set G = g1 union g2;
the carrier of (X1 union X2) = the carrier of X1 \/ the carrier of X2 by TSEP_1:def_2;
then reconsider F = g1 union g2 as Function of (X1 union X2),Y ;
take F ; ::_thesis: ( F | X1 = f1 & F | X2 = f2 )
( (g1 union g2) | A1 = f1 & (g1 union g2) | A2 = f2 ) by A4, Def1, Th1;
hence ( F | X1 = f1 & F | X2 = f2 ) by A2, Def5; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (X1 union X2),Y st b1 | X1 = f1 & b1 | X2 = f2 & b2 | X1 = f1 & b2 | X2 = f2 holds
b1 = b2
proof
set A = the carrier of (X1 union X2);
A9: X2 is SubSpace of X1 union X2 by TSEP_1:22;
set A2 = the carrier of X2;
A10: X1 is SubSpace of X1 union X2 by TSEP_1:22;
set A1 = the carrier of X1;
let F, G be Function of (X1 union X2),Y; ::_thesis: ( F | X1 = f1 & F | X2 = f2 & G | X1 = f1 & G | X2 = f2 implies F = G )
assume that
A11: F | X1 = f1 and
A12: F | X2 = f2 and
A13: G | X1 = f1 and
A14: G | X2 = f2 ; ::_thesis: F = G
A15: the carrier of (X1 union X2) = the carrier of X1 \/ the carrier of X2 by TSEP_1:def_2;
now__::_thesis:_for_a_being_Element_of_the_carrier_of_(X1_union_X2)_holds_F_._a_=_G_._a
let a be Element of the carrier of (X1 union X2); ::_thesis: F . a = G . a
A16: now__::_thesis:_(_a_in_the_carrier_of_X2_implies_F_._a_=_G_._a_)
assume A17: a in the carrier of X2 ; ::_thesis: F . a = G . a
hence F . a = (F | the carrier of X2) . a by FUNCT_1:49
.= f2 . a by A12, A9, Def5
.= (G | the carrier of X2) . a by A14, A9, Def5
.= G . a by A17, FUNCT_1:49 ;
::_thesis: verum
end;
now__::_thesis:_(_a_in_the_carrier_of_X1_implies_F_._a_=_G_._a_)
assume A18: a in the carrier of X1 ; ::_thesis: F . a = G . a
hence F . a = (F | the carrier of X1) . a by FUNCT_1:49
.= f1 . a by A11, A10, Def5
.= (G | the carrier of X1) . a by A13, A10, Def5
.= G . a by A18, FUNCT_1:49 ;
::_thesis: verum
end;
hence F . a = G . a by A15, A16, XBOOLE_0:def_3; ::_thesis: verum
end;
hence F = G by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def12 defines union TMAP_1:def_12_:_
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for f1 being Function of X1,Y
for f2 being Function of X2,Y st ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) holds
for b7 being Function of (X1 union X2),Y holds
( b7 = f1 union f2 iff ( b7 | X1 = f1 & b7 | X2 = f2 ) );
theorem Th126: :: TMAP_1:126
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for g being Function of (X1 union X2),Y holds g = (g | X1) union (g | X2)
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for g being Function of (X1 union X2),Y holds g = (g | X1) union (g | X2)
let X1, X2 be non empty SubSpace of X; ::_thesis: for g being Function of (X1 union X2),Y holds g = (g | X1) union (g | X2)
let g be Function of (X1 union X2),Y; ::_thesis: g = (g | X1) union (g | X2)
now__::_thesis:_(_X1_meets_X2_implies_(g_|_X1)_|_(X1_meet_X2)_=_(g_|_X2)_|_(X1_meet_X2)_)
assume A1: X1 meets X2 ; ::_thesis: (g | X1) | (X1 meet X2) = (g | X2) | (X1 meet X2)
then A2: X1 meet X2 is SubSpace of X2 by TSEP_1:27;
A3: X2 is SubSpace of X1 union X2 by TSEP_1:22;
A4: X1 is SubSpace of X1 union X2 by TSEP_1:22;
X1 meet X2 is SubSpace of X1 by A1, TSEP_1:27;
hence (g | X1) | (X1 meet X2) = g | (X1 meet X2) by A4, Th72
.= (g | X2) | (X1 meet X2) by A2, A3, Th72 ;
::_thesis: verum
end;
hence g = (g | X1) union (g | X2) by Def12; ::_thesis: verum
end;
theorem :: TMAP_1:127
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
for g being Function of X,Y holds g = (g | X1) union (g | X2)
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
for g being Function of X,Y holds g = (g | X1) union (g | X2)
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 implies for g being Function of X,Y holds g = (g | X1) union (g | X2) )
assume A1: X = X1 union X2 ; ::_thesis: for g being Function of X,Y holds g = (g | X1) union (g | X2)
let g be Function of X,Y; ::_thesis: g = (g | X1) union (g | X2)
reconsider h = g as Function of (X1 union X2),Y by A1;
X2 is SubSpace of X1 union X2 by TSEP_1:22;
then A2: h | X2 = g | X2 by Def5;
X1 is SubSpace of X1 union X2 by TSEP_1:22;
then h | X1 = g | X1 by Def5;
hence g = (g | X1) union (g | X2) by A2, Th126; ::_thesis: verum
end;
theorem Th128: :: TMAP_1:128
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
for f1 being Function of X1,Y
for f2 being Function of X2,Y holds
( ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) iff f1 | (X1 meet X2) = f2 | (X1 meet X2) )
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
for f1 being Function of X1,Y
for f2 being Function of X2,Y holds
( ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) iff f1 | (X1 meet X2) = f2 | (X1 meet X2) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies for f1 being Function of X1,Y
for f2 being Function of X2,Y holds
( ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) iff f1 | (X1 meet X2) = f2 | (X1 meet X2) ) )
assume A1: X1 meets X2 ; ::_thesis: for f1 being Function of X1,Y
for f2 being Function of X2,Y holds
( ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) iff f1 | (X1 meet X2) = f2 | (X1 meet X2) )
let f1 be Function of X1,Y; ::_thesis: for f2 being Function of X2,Y holds
( ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) iff f1 | (X1 meet X2) = f2 | (X1 meet X2) )
let f2 be Function of X2,Y; ::_thesis: ( ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) iff f1 | (X1 meet X2) = f2 | (X1 meet X2) )
thus ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 implies f1 | (X1 meet X2) = f2 | (X1 meet X2) ) ::_thesis: ( f1 | (X1 meet X2) = f2 | (X1 meet X2) implies ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) )
proof
A2: ( X1 meet X2 is SubSpace of X2 & X2 is SubSpace of X1 union X2 ) by A1, TSEP_1:22, TSEP_1:27;
assume that
A3: (f1 union f2) | X1 = f1 and
A4: (f1 union f2) | X2 = f2 ; ::_thesis: f1 | (X1 meet X2) = f2 | (X1 meet X2)
( X1 meet X2 is SubSpace of X1 & X1 is SubSpace of X1 union X2 ) by A1, TSEP_1:22, TSEP_1:27;
then (f1 union f2) | (X1 meet X2) = f1 | (X1 meet X2) by A3, Th72;
hence f1 | (X1 meet X2) = f2 | (X1 meet X2) by A2, A4, Th72; ::_thesis: verum
end;
thus ( f1 | (X1 meet X2) = f2 | (X1 meet X2) implies ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) ) by Def12; ::_thesis: verum
end;
theorem :: TMAP_1:129
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for f1 being Function of X1,Y
for f2 being Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
( ( X1 is SubSpace of X2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies X1 is SubSpace of X2 ) & ( X2 is SubSpace of X1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies X2 is SubSpace of X1 ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for f1 being Function of X1,Y
for f2 being Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
( ( X1 is SubSpace of X2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies X1 is SubSpace of X2 ) & ( X2 is SubSpace of X1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies X2 is SubSpace of X1 ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: for f1 being Function of X1,Y
for f2 being Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
( ( X1 is SubSpace of X2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies X1 is SubSpace of X2 ) & ( X2 is SubSpace of X1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies X2 is SubSpace of X1 ) )
let f1 be Function of X1,Y; ::_thesis: for f2 being Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
( ( X1 is SubSpace of X2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies X1 is SubSpace of X2 ) & ( X2 is SubSpace of X1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies X2 is SubSpace of X1 ) )
let f2 be Function of X2,Y; ::_thesis: ( f1 | (X1 meet X2) = f2 | (X1 meet X2) implies ( ( X1 is SubSpace of X2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies X1 is SubSpace of X2 ) & ( X2 is SubSpace of X1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies X2 is SubSpace of X1 ) ) )
reconsider Y1 = X1, Y2 = X2, Y3 = X1 union X2 as SubSpace of X1 union X2 by TSEP_1:2, TSEP_1:22;
assume A1: f1 | (X1 meet X2) = f2 | (X1 meet X2) ; ::_thesis: ( ( X1 is SubSpace of X2 implies f1 union f2 = f2 ) & ( f1 union f2 = f2 implies X1 is SubSpace of X2 ) & ( X2 is SubSpace of X1 implies f1 union f2 = f1 ) & ( f1 union f2 = f1 implies X2 is SubSpace of X1 ) )
A2: now__::_thesis:_(_X1_is_SubSpace_of_X2_implies_f1_union_f2_=_f2_)
assume X1 is SubSpace of X2 ; ::_thesis: f1 union f2 = f2
then A3: TopStruct(# the carrier of X2, the topology of X2 #) = X1 union X2 by TSEP_1:23;
(f1 union f2) | X2 = f2 by A1, Def12;
then (f1 union f2) | the carrier of Y2 = f2 by Def5;
then (f1 union f2) | the carrier of Y3 = f2 by A3;
then (f1 union f2) | (X1 union X2) = f2 by Def5;
hence f1 union f2 = f2 by Th67; ::_thesis: verum
end;
A4: now__::_thesis:_(_X2_is_SubSpace_of_X1_implies_f1_union_f2_=_f1_)
assume X2 is SubSpace of X1 ; ::_thesis: f1 union f2 = f1
then A5: TopStruct(# the carrier of X1, the topology of X1 #) = X1 union X2 by TSEP_1:23;
(f1 union f2) | X1 = f1 by A1, Def12;
then (f1 union f2) | the carrier of Y1 = f1 by Def5;
then (f1 union f2) | the carrier of Y3 = f1 by A5;
then (f1 union f2) | (X1 union X2) = f1 by Def5;
hence f1 union f2 = f1 by Th67; ::_thesis: verum
end;
now__::_thesis:_(_f1_union_f2_=_f2_implies_X1_is_SubSpace_of_X2_)
A6: ( dom (f1 union f2) = the carrier of (X1 union X2) & dom f2 = the carrier of X2 ) by FUNCT_2:def_1;
assume f1 union f2 = f2 ; ::_thesis: X1 is SubSpace of X2
then X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) by A6, TSEP_1:5;
hence X1 is SubSpace of X2 by TSEP_1:23; ::_thesis: verum
end;
hence ( X1 is SubSpace of X2 iff f1 union f2 = f2 ) by A2; ::_thesis: ( X2 is SubSpace of X1 iff f1 union f2 = f1 )
now__::_thesis:_(_f1_union_f2_=_f1_implies_X2_is_SubSpace_of_X1_)
A7: ( dom (f1 union f2) = the carrier of (X1 union X2) & dom f1 = the carrier of X1 ) by FUNCT_2:def_1;
assume f1 union f2 = f1 ; ::_thesis: X2 is SubSpace of X1
then X1 union X2 = TopStruct(# the carrier of X1, the topology of X1 #) by A7, TSEP_1:5;
hence X2 is SubSpace of X1 by TSEP_1:23; ::_thesis: verum
end;
hence ( X2 is SubSpace of X1 iff f1 union f2 = f1 ) by A4; ::_thesis: verum
end;
theorem :: TMAP_1:130
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for f1 being Function of X1,Y
for f2 being Function of X2,Y st ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) holds
f1 union f2 = f2 union f1
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for f1 being Function of X1,Y
for f2 being Function of X2,Y st ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) holds
f1 union f2 = f2 union f1
let X1, X2 be non empty SubSpace of X; ::_thesis: for f1 being Function of X1,Y
for f2 being Function of X2,Y st ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) holds
f1 union f2 = f2 union f1
let f1 be Function of X1,Y; ::_thesis: for f2 being Function of X2,Y st ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) holds
f1 union f2 = f2 union f1
let f2 be Function of X2,Y; ::_thesis: ( ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) implies f1 union f2 = f2 union f1 )
assume ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) ; ::_thesis: f1 union f2 = f2 union f1
then ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) by Def12;
hence f1 union f2 = f2 union f1 by Th126; ::_thesis: verum
end;
theorem :: TMAP_1:131
for X, Y being non empty TopSpace
for X1, X2, X3 being non empty SubSpace of X
for f1 being Function of X1,Y
for f2 being Function of X2,Y
for f3 being Function of X3,Y st ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) & ( X1 misses X3 or f1 | (X1 meet X3) = f3 | (X1 meet X3) ) & ( X2 misses X3 or f2 | (X2 meet X3) = f3 | (X2 meet X3) ) holds
(f1 union f2) union f3 = f1 union (f2 union f3)
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2, X3 being non empty SubSpace of X
for f1 being Function of X1,Y
for f2 being Function of X2,Y
for f3 being Function of X3,Y st ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) & ( X1 misses X3 or f1 | (X1 meet X3) = f3 | (X1 meet X3) ) & ( X2 misses X3 or f2 | (X2 meet X3) = f3 | (X2 meet X3) ) holds
(f1 union f2) union f3 = f1 union (f2 union f3)
let X1, X2, X3 be non empty SubSpace of X; ::_thesis: for f1 being Function of X1,Y
for f2 being Function of X2,Y
for f3 being Function of X3,Y st ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) & ( X1 misses X3 or f1 | (X1 meet X3) = f3 | (X1 meet X3) ) & ( X2 misses X3 or f2 | (X2 meet X3) = f3 | (X2 meet X3) ) holds
(f1 union f2) union f3 = f1 union (f2 union f3)
let f1 be Function of X1,Y; ::_thesis: for f2 being Function of X2,Y
for f3 being Function of X3,Y st ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) & ( X1 misses X3 or f1 | (X1 meet X3) = f3 | (X1 meet X3) ) & ( X2 misses X3 or f2 | (X2 meet X3) = f3 | (X2 meet X3) ) holds
(f1 union f2) union f3 = f1 union (f2 union f3)
let f2 be Function of X2,Y; ::_thesis: for f3 being Function of X3,Y st ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) & ( X1 misses X3 or f1 | (X1 meet X3) = f3 | (X1 meet X3) ) & ( X2 misses X3 or f2 | (X2 meet X3) = f3 | (X2 meet X3) ) holds
(f1 union f2) union f3 = f1 union (f2 union f3)
let f3 be Function of X3,Y; ::_thesis: ( ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) & ( X1 misses X3 or f1 | (X1 meet X3) = f3 | (X1 meet X3) ) & ( X2 misses X3 or f2 | (X2 meet X3) = f3 | (X2 meet X3) ) implies (f1 union f2) union f3 = f1 union (f2 union f3) )
assume that
A1: ( X1 misses X2 or f1 | (X1 meet X2) = f2 | (X1 meet X2) ) and
A2: ( X1 misses X3 or f1 | (X1 meet X3) = f3 | (X1 meet X3) ) and
A3: ( X2 misses X3 or f2 | (X2 meet X3) = f3 | (X2 meet X3) ) ; ::_thesis: (f1 union f2) union f3 = f1 union (f2 union f3)
set g = (f1 union f2) union f3;
A4: (X1 union X2) union X3 = X1 union (X2 union X3) by TSEP_1:21;
then reconsider f = (f1 union f2) union f3 as Function of (X1 union (X2 union X3)),Y ;
A5: X1 union X2 is SubSpace of X1 union (X2 union X3) by A4, TSEP_1:22;
A6: now__::_thesis:_(_X1_union_X2_meets_X3_implies_(f1_union_f2)_|_((X1_union_X2)_meet_X3)_=_f3_|_((X1_union_X2)_meet_X3)_)
assume A7: X1 union X2 meets X3 ; ::_thesis: (f1 union f2) | ((X1 union X2) meet X3) = f3 | ((X1 union X2) meet X3)
now__::_thesis:_(f1_union_f2)_|_((X1_union_X2)_meet_X3)_=_f3_|_((X1_union_X2)_meet_X3)
percases ( ( X1 meets X3 & not X2 meets X3 ) or ( not X1 meets X3 & X2 meets X3 ) or ( X1 meets X3 & X2 meets X3 ) ) by A7, Th34;
supposeA8: ( X1 meets X3 & not X2 meets X3 ) ; ::_thesis: (f1 union f2) | ((X1 union X2) meet X3) = f3 | ((X1 union X2) meet X3)
then A9: (X1 union X2) meet X3 = X1 meet X3 by Th26;
A10: X1 is SubSpace of X1 union X2 by TSEP_1:22;
X1 meet X3 is SubSpace of X1 by A8, TSEP_1:27;
then (f1 union f2) | (X1 meet X3) = ((f1 union f2) | X1) | (X1 meet X3) by A10, Th72
.= f1 | (X1 meet X3) by A1, Def12 ;
hence (f1 union f2) | ((X1 union X2) meet X3) = f3 | ((X1 union X2) meet X3) by A2, A8, A9; ::_thesis: verum
end;
supposeA11: ( not X1 meets X3 & X2 meets X3 ) ; ::_thesis: (f1 union f2) | ((X1 union X2) meet X3) = f3 | ((X1 union X2) meet X3)
then A12: (X1 union X2) meet X3 = X2 meet X3 by Th26;
A13: X2 is SubSpace of X1 union X2 by TSEP_1:22;
X2 meet X3 is SubSpace of X2 by A11, TSEP_1:27;
then (f1 union f2) | (X2 meet X3) = ((f1 union f2) | X2) | (X2 meet X3) by A13, Th72
.= f2 | (X2 meet X3) by A1, Def12 ;
hence (f1 union f2) | ((X1 union X2) meet X3) = f3 | ((X1 union X2) meet X3) by A3, A11, A12; ::_thesis: verum
end;
supposeA14: ( X1 meets X3 & X2 meets X3 ) ; ::_thesis: (f1 union f2) | ((X1 union X2) meet X3) = f3 | ((X1 union X2) meet X3)
then ( X1 meet X3 is SubSpace of X3 & X2 meet X3 is SubSpace of X3 ) by TSEP_1:27;
then A15: (X1 meet X3) union (X2 meet X3) is SubSpace of X3 by Th24;
A16: X2 meet X3 is SubSpace of X2 by A14, TSEP_1:27;
A17: X1 meet X3 is SubSpace of (X1 meet X3) union (X2 meet X3) by TSEP_1:22;
then A18: (f3 | ((X1 meet X3) union (X2 meet X3))) | (X1 meet X3) = f3 | (X1 meet X3) by A15, Th72;
A19: X1 meet X3 is SubSpace of X1 by A14, TSEP_1:27;
then A20: (X1 meet X3) union (X2 meet X3) is SubSpace of X1 union X2 by A16, Th22;
then A21: ((f1 union f2) | ((X1 meet X3) union (X2 meet X3))) | (X1 meet X3) = (f1 union f2) | (X1 meet X3) by A17, Th72;
X2 is SubSpace of X1 union X2 by TSEP_1:22;
then A22: (f1 union f2) | (X2 meet X3) = ((f1 union f2) | X2) | (X2 meet X3) by A16, Th72
.= f2 | (X2 meet X3) by A1, Def12 ;
set v = f3 | ((X1 meet X3) union (X2 meet X3));
A23: X2 meet X3 is SubSpace of (X1 meet X3) union (X2 meet X3) by TSEP_1:22;
then A24: (f3 | ((X1 meet X3) union (X2 meet X3))) | (X2 meet X3) = f3 | (X2 meet X3) by A15, Th72;
X1 is SubSpace of X1 union X2 by TSEP_1:22;
then A25: (f1 union f2) | (X1 meet X3) = ((f1 union f2) | X1) | (X1 meet X3) by A19, Th72
.= f1 | (X1 meet X3) by A1, Def12 ;
A26: ((f1 union f2) | ((X1 meet X3) union (X2 meet X3))) | (X2 meet X3) = (f1 union f2) | (X2 meet X3) by A20, A23, Th72;
(f1 union f2) | ((X1 union X2) meet X3) = (f1 union f2) | ((X1 meet X3) union (X2 meet X3)) by A14, TSEP_1:32
.= ((f3 | ((X1 meet X3) union (X2 meet X3))) | (X1 meet X3)) union ((f3 | ((X1 meet X3) union (X2 meet X3))) | (X2 meet X3)) by A2, A3, A14, A25, A22, A21, A26, A18, A24, Th126
.= f3 | ((X1 meet X3) union (X2 meet X3)) by Th126 ;
hence (f1 union f2) | ((X1 union X2) meet X3) = f3 | ((X1 union X2) meet X3) by A14, TSEP_1:32; ::_thesis: verum
end;
end;
end;
hence (f1 union f2) | ((X1 union X2) meet X3) = f3 | ((X1 union X2) meet X3) ; ::_thesis: verum
end;
then ( X1 union X2 is SubSpace of (X1 union X2) union X3 & ((f1 union f2) union f3) | (X1 union X2) = f1 union f2 ) by Def12, TSEP_1:22;
then A27: f | the carrier of (X1 union X2) = f1 union f2 by Def5;
A28: X3 is SubSpace of X1 union (X2 union X3) by A4, TSEP_1:22;
A29: X2 union X3 is SubSpace of X1 union (X2 union X3) by TSEP_1:22;
( X3 is SubSpace of (X1 union X2) union X3 & ((f1 union f2) union f3) | X3 = f3 ) by A6, Def12, TSEP_1:22;
then A30: f | the carrier of X3 = f3 by Def5;
A31: X1 union X2 is SubSpace of X1 union (X2 union X3) by A4, TSEP_1:22;
X3 is SubSpace of X2 union X3 by TSEP_1:22;
then A32: (f | (X2 union X3)) | X3 = f | X3 by A29, Th72
.= f3 by A28, A30, Def5 ;
X2 is SubSpace of X1 union X2 by TSEP_1:22;
then A33: f | X2 = (f | (X1 union X2)) | X2 by A31, Th72
.= (f1 union f2) | X2 by A5, A27, Def5 ;
X2 is SubSpace of X2 union X3 by TSEP_1:22;
then (f | (X2 union X3)) | X2 = f | X2 by A29, Th72
.= f2 by A1, A33, Def12 ;
then A34: f | (X2 union X3) = f2 union f3 by A32, Th126;
X1 is SubSpace of X1 union X2 by TSEP_1:22;
then f | X1 = (f | (X1 union X2)) | X1 by A31, Th72
.= (f1 union f2) | X1 by A5, A27, Def5 ;
then f | X1 = f1 by A1, Def12;
hence (f1 union f2) union f3 = f1 union (f2 union f3) by A34, Th126; ::_thesis: verum
end;
theorem :: TMAP_1:132
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) & X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of (X1 union X2),Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) & X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of (X1 union X2),Y
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) & X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of (X1 union X2),Y )
assume A1: X1 meets X2 ; ::_thesis: for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) & X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of (X1 union X2),Y
let f1 be continuous Function of X1,Y; ::_thesis: for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) & X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of (X1 union X2),Y
let f2 be continuous Function of X2,Y; ::_thesis: ( f1 | (X1 meet X2) = f2 | (X1 meet X2) & X1,X2 are_weakly_separated implies f1 union f2 is continuous Function of (X1 union X2),Y )
assume f1 | (X1 meet X2) = f2 | (X1 meet X2) ; ::_thesis: ( not X1,X2 are_weakly_separated or f1 union f2 is continuous Function of (X1 union X2),Y )
then A2: ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) by A1, Th128;
assume X1,X2 are_weakly_separated ; ::_thesis: f1 union f2 is continuous Function of (X1 union X2),Y
hence f1 union f2 is continuous Function of (X1 union X2),Y by A2, Th114; ::_thesis: verum
end;
theorem Th133: :: TMAP_1:133
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 misses X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of (X1 union X2),Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 misses X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of (X1 union X2),Y
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 misses X2 implies for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of (X1 union X2),Y )
assume A1: X1 misses X2 ; ::_thesis: for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of (X1 union X2),Y
let f1 be continuous Function of X1,Y; ::_thesis: for f2 being continuous Function of X2,Y st X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of (X1 union X2),Y
let f2 be continuous Function of X2,Y; ::_thesis: ( X1,X2 are_weakly_separated implies f1 union f2 is continuous Function of (X1 union X2),Y )
assume A2: X1,X2 are_weakly_separated ; ::_thesis: f1 union f2 is continuous Function of (X1 union X2),Y
( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) by A1, Def12;
hence f1 union f2 is continuous Function of (X1 union X2),Y by A2, Th114; ::_thesis: verum
end;
theorem :: TMAP_1:134
for X, Y being non empty TopSpace
for X1, X2 being non empty closed SubSpace of X st X1 meets X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty closed SubSpace of X st X1 meets X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
let X1, X2 be non empty closed SubSpace of X; ::_thesis: ( X1 meets X2 implies for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y )
assume A1: X1 meets X2 ; ::_thesis: for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
let f1 be continuous Function of X1,Y; ::_thesis: for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
let f2 be continuous Function of X2,Y; ::_thesis: ( f1 | (X1 meet X2) = f2 | (X1 meet X2) implies f1 union f2 is continuous Function of (X1 union X2),Y )
assume f1 | (X1 meet X2) = f2 | (X1 meet X2) ; ::_thesis: f1 union f2 is continuous Function of (X1 union X2),Y
then ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) by A1, Th128;
hence f1 union f2 is continuous Function of (X1 union X2),Y by Th115; ::_thesis: verum
end;
theorem :: TMAP_1:135
for X, Y being non empty TopSpace
for X1, X2 being non empty open SubSpace of X st X1 meets X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty open SubSpace of X st X1 meets X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
let X1, X2 be non empty open SubSpace of X; ::_thesis: ( X1 meets X2 implies for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y )
assume A1: X1 meets X2 ; ::_thesis: for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
let f1 be continuous Function of X1,Y; ::_thesis: for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
let f2 be continuous Function of X2,Y; ::_thesis: ( f1 | (X1 meet X2) = f2 | (X1 meet X2) implies f1 union f2 is continuous Function of (X1 union X2),Y )
assume f1 | (X1 meet X2) = f2 | (X1 meet X2) ; ::_thesis: f1 union f2 is continuous Function of (X1 union X2),Y
then ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) by A1, Th128;
hence f1 union f2 is continuous Function of (X1 union X2),Y by Th116; ::_thesis: verum
end;
theorem :: TMAP_1:136
for X, Y being non empty TopSpace
for X1, X2 being non empty closed SubSpace of X st X1 misses X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty closed SubSpace of X st X1 misses X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y
let X1, X2 be non empty closed SubSpace of X; ::_thesis: ( X1 misses X2 implies for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y )
assume A1: X1 misses X2 ; ::_thesis: for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y
let f1 be continuous Function of X1,Y; ::_thesis: for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y
let f2 be continuous Function of X2,Y; ::_thesis: f1 union f2 is continuous Function of (X1 union X2),Y
( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) by A1, Def12;
hence f1 union f2 is continuous Function of (X1 union X2),Y by Th115; ::_thesis: verum
end;
theorem :: TMAP_1:137
for X, Y being non empty TopSpace
for X1, X2 being non empty open SubSpace of X st X1 misses X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty open SubSpace of X st X1 misses X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y
let X1, X2 be non empty open SubSpace of X; ::_thesis: ( X1 misses X2 implies for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y )
assume A1: X1 misses X2 ; ::_thesis: for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y
let f1 be continuous Function of X1,Y; ::_thesis: for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y
let f2 be continuous Function of X2,Y; ::_thesis: f1 union f2 is continuous Function of (X1 union X2),Y
( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) by A1, Def12;
hence f1 union f2 is continuous Function of (X1 union X2),Y by Th116; ::_thesis: verum
end;
theorem :: TMAP_1:138
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_separated iff ( X1 misses X2 & ( for Y being non empty TopSpace
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y ) ) )
thus ( X1,X2 are_separated implies ( X1 misses X2 & ( for Y being non empty TopSpace
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y ) ) ) ::_thesis: ( X1 misses X2 & ( for Y being non empty TopSpace
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y ) implies X1,X2 are_separated )
proof
assume A1: X1,X2 are_separated ; ::_thesis: ( X1 misses X2 & ( for Y being non empty TopSpace
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y ) )
hence X1 misses X2 by TSEP_1:63; ::_thesis: for Y being non empty TopSpace
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y
X1,X2 are_weakly_separated by A1, TSEP_1:78;
hence for Y being non empty TopSpace
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y by A1, Th133, TSEP_1:63; ::_thesis: verum
end;
thus ( X1 misses X2 & ( for Y being non empty TopSpace
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y ) implies X1,X2 are_separated ) ::_thesis: verum
proof
assume A2: X1 misses X2 ; ::_thesis: ( ex Y being non empty TopSpace ex f1 being continuous Function of X1,Y ex f2 being continuous Function of X2,Y st f1 union f2 is not continuous Function of (X1 union X2),Y or X1,X2 are_separated )
assume A3: for Y being non empty TopSpace
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y holds f1 union f2 is continuous Function of (X1 union X2),Y ; ::_thesis: X1,X2 are_separated
now__::_thesis:_for_Y_being_non_empty_TopSpace
for_g_being_Function_of_(X1_union_X2),Y_st_g_|_X1_is_continuous_Function_of_X1,Y_&_g_|_X2_is_continuous_Function_of_X2,Y_holds_
g_is_continuous_Function_of_(X1_union_X2),Y
let Y be non empty TopSpace; ::_thesis: for g being Function of (X1 union X2),Y st g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y holds
g is continuous Function of (X1 union X2),Y
let g be Function of (X1 union X2),Y; ::_thesis: ( g | X1 is continuous Function of X1,Y & g | X2 is continuous Function of X2,Y implies g is continuous Function of (X1 union X2),Y )
assume that
A4: g | X1 is continuous Function of X1,Y and
A5: g | X2 is continuous Function of X2,Y ; ::_thesis: g is continuous Function of (X1 union X2),Y
reconsider f2 = g | X2 as continuous Function of X2,Y by A5;
reconsider f1 = g | X1 as continuous Function of X1,Y by A4;
g = f1 union f2 by Th126;
hence g is continuous Function of (X1 union X2),Y by A3; ::_thesis: verum
end;
hence X1,X2 are_separated by A2, Th123; ::_thesis: verum
end;
end;
theorem :: TMAP_1:139
for X, Y being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 & X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of X,Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 & X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of X,Y
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 implies for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 & X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of X,Y )
assume A1: X = X1 union X2 ; ::_thesis: for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 & X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of X,Y
let f1 be continuous Function of X1,Y; ::_thesis: for f2 being continuous Function of X2,Y st (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 & X1,X2 are_weakly_separated holds
f1 union f2 is continuous Function of X,Y
let f2 be continuous Function of X2,Y; ::_thesis: ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 & X1,X2 are_weakly_separated implies f1 union f2 is continuous Function of X,Y )
assume A2: ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) ; ::_thesis: ( not X1,X2 are_weakly_separated or f1 union f2 is continuous Function of X,Y )
reconsider g = f1 union f2 as Function of X,Y by A1;
assume A3: X1,X2 are_weakly_separated ; ::_thesis: f1 union f2 is continuous Function of X,Y
( g | X1 = f1 & g | X2 = f2 ) by A1, A2, Def5;
hence f1 union f2 is continuous Function of X,Y by A1, A3, Th120; ::_thesis: verum
end;
theorem :: TMAP_1:140
for X, Y being non empty TopSpace
for X1, X2 being non empty closed SubSpace of X
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty closed SubSpace of X
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
let X1, X2 be non empty closed SubSpace of X; ::_thesis: for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
let f1 be continuous Function of X1,Y; ::_thesis: for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
let f2 be continuous Function of X2,Y; ::_thesis: ( X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 implies f1 union f2 is continuous Function of X,Y )
assume that
A1: X = X1 union X2 and
A2: ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) ; ::_thesis: f1 union f2 is continuous Function of X,Y
reconsider g = f1 union f2 as Function of X,Y by A1;
( g | X1 = f1 & g | X2 = f2 ) by A1, A2, Def5;
hence f1 union f2 is continuous Function of X,Y by A1, Th121; ::_thesis: verum
end;
theorem :: TMAP_1:141
for X, Y being non empty TopSpace
for X1, X2 being non empty open SubSpace of X
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
proof
let X, Y be non empty TopSpace; ::_thesis: for X1, X2 being non empty open SubSpace of X
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
let X1, X2 be non empty open SubSpace of X; ::_thesis: for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
let f1 be continuous Function of X1,Y; ::_thesis: for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
let f2 be continuous Function of X2,Y; ::_thesis: ( X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 implies f1 union f2 is continuous Function of X,Y )
assume that
A1: X = X1 union X2 and
A2: ( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 ) ; ::_thesis: f1 union f2 is continuous Function of X,Y
reconsider g = f1 union f2 as Function of X,Y by A1;
( g | X1 = f1 & g | X2 = f2 ) by A1, A2, Def5;
hence f1 union f2 is continuous Function of X,Y by A1, Th122; ::_thesis: verum
end;