:: JORDAN13 semantic presentation

begin


definition
let C be non empty non horizontal non vertical being_simple_closed_curve Subset of (TOP-REAL 2);
let n be Element of NAT ;
assume A1: n is_sufficiently_large_for C ;
func Span (C,n) ->  non constant standard clockwise_oriented special_circular_sequence means :: JORDAN13:def 1
( it is_sequence_on Gauge (C,n) & it /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) & it /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Element of NAT st 1 <= k & k + 2 <= len it holds
( ( front_right_cell (it,k,(Gauge (C,n))) misses C & front_left_cell (it,k,(Gauge (C,n))) misses C implies it turns_left k, Gauge (C,n) ) & ( front_right_cell (it,k,(Gauge (C,n))) misses C & front_left_cell (it,k,(Gauge (C,n))) meets C implies it goes_straight k, Gauge (C,n) ) & ( front_right_cell (it,k,(Gauge (C,n))) meets C implies it turns_right k, Gauge (C,n) ) ) ) );
existence
 ex b1 being non constant standard clockwise_oriented special_circular_sequence st
( b1 is_sequence_on Gauge (C,n) & b1 /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) & b1 /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Element of NAT st 1 <= k & k + 2 <= len b1 holds
( ( front_right_cell (b1,k,(Gauge (C,n))) misses C & front_left_cell (b1,k,(Gauge (C,n))) misses C implies b1 turns_left k, Gauge (C,n) ) & ( front_right_cell (b1,k,(Gauge (C,n))) misses C & front_left_cell (b1,k,(Gauge (C,n))) meets C implies b1 goes_straight k, Gauge (C,n) ) & ( front_right_cell (b1,k,(Gauge (C,n))) meets C implies b1 turns_right k, Gauge (C,n) ) ) ) )
proof
set XS = X-SpanStart (C,n);
set YS = Y-SpanStart (C,n);
set G = Gauge (C,n);
A2: len (Gauge (C,n)) = (2 |^ n) + 3 by JORDAN8:def_1;
defpred S1[ Element of NAT , set , set ] means ( ( $1 = 0 implies $3 = <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))))*> ) & ( $1 = 1 implies $3 = <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n)))),((Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))))*> ) & ( $1 > 1 & $2 is FinSequence of (TOP-REAL 2) implies for f being FinSequence of (TOP-REAL 2) st $2 = f holds
( ( len f = $1 implies ( ( f is_sequence_on Gauge (C,n) & left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C implies ( ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len f) -' 1, Gauge (C,n) & $3 = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) & ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len f) -' 1, Gauge (C,n) & $3 = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) & ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len f) -' 1, Gauge (C,n) & $3 = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) ) ) & ( ( not f is_sequence_on Gauge (C,n) or left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) implies $3 = f ^ <*((Gauge (C,n)) * (1,1))*> ) ) ) & ( len f <> $1 implies $3 = {} ) ) ) & ( $1 > 1 & $2 is not FinSequence of (TOP-REAL 2) implies $3 = {} ) );
A3: 1 + 1 <= X-SpanStart (C,n) by JORDAN1H:49;
A4: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8;
A5: [(X-SpanStart (C,n)),(Y-SpanStart (C,n))] in Indices (Gauge (C,n)) by A1, JORDAN11:8;
A6: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A7: for k being Element of NAT
for x being set ex y being set st S1[k,x,y]
proof
let k be Element of NAT ; ::_thesis: for x being set ex y being set st S1[k,x,y]
let x be set ; ::_thesis: ex y being set st S1[k,x,y]
percases ( k = 0 or k = 1 or ( k > 1 & x is FinSequence of (TOP-REAL 2) ) or ( k > 1 & x is not FinSequence of (TOP-REAL 2) ) ) by NAT_1:25;
supposeA8: k = 0 ; ::_thesis: ex y being set st S1[k,x,y]
take  <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))))*> ; ::_thesis: S1[k,x,<*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))))*>]
thus  S1[k,x,<*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))))*>] by A8; ::_thesis: verum
end;
supposeA9: k = 1 ; ::_thesis: ex y being set st S1[k,x,y]
take  <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n)))),((Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))))*> ; ::_thesis: S1[k,x,<*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n)))),((Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))))*>]
thus  S1[k,x,<*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n)))),((Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))))*>] by A9; ::_thesis: verum
end;
supposethat A10: k > 1 and
A11: x is FinSequence of (TOP-REAL 2) ; ::_thesis: ex y being set st S1[k,x,y]
reconsider f = x as FinSequence of (TOP-REAL 2) by A11;
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
percases ( len f = k or len f <> k ) ;
supposeA12: len f = k ; ::_thesis: ex y being set st S1[k,x,y]
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
percases ( ( f is_sequence_on Gauge (C,n) & left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) or not f is_sequence_on Gauge (C,n) or left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) ;
supposeA13: ( f is_sequence_on Gauge (C,n) & left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ex y being set st S1[k,x,y]
A14: ((len f) -' 1) + 1 = len f by A10, A12, XREAL_1:235;
then A15: ((len f) -' 1) + (1 + 1) = (len f) + 1 ;
A16: ((len f) -' 1) + 1 in dom f by A10, A12, A14, FINSEQ_3:25;
A17: 1 <= (len f) -' 1 by A10, A12, NAT_D:49;
then consider i1, j1, i2, j2 being Element of NAT  such that
A18: [i1,j1] in Indices (Gauge (C,n)) and
A19: f /. ((len f) -' 1) = (Gauge (C,n)) * (i1,j1) and
A20: [i2,j2] in Indices (Gauge (C,n)) and
A21: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i2,j2) and
A22: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A13, A14, JORDAN8:3;
A23: i1 <= len (Gauge (C,n)) by A18, MATRIX_1:38;
A24: 1 <= j2 + 1 by NAT_1:12;
A25: 1 <= i2 by A20, MATRIX_1:38;
A26: j1 <= width (Gauge (C,n)) by A18, MATRIX_1:38;
A27: 1 <= i2 + 1 by NAT_1:12;
A28: 1 <= j2 by A20, MATRIX_1:38;
 (len f) -' 1 <= len f by NAT_D:35;
then A29: (len f) -' 1 in dom f by A17, FINSEQ_3:25;
A30: j2 <= width (Gauge (C,n)) by A20, MATRIX_1:38;
then A31: j2 -' 1 <= width (Gauge (C,n)) by NAT_D:44;
A32: i2 <= len (Gauge (C,n)) by A20, MATRIX_1:38;
then A33: i2 -' 1 <= len (Gauge (C,n)) by NAT_D:44;
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
percases ( ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) or ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) or front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) ;
supposeA34: ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ex y being set st S1[k,x,y]
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A22;
supposeA35: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_turns_left_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2 -' 1; ::_thesis: ex j being Element of NAT st
( f1 turns_left (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2; ::_thesis: ( f1 turns_left (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)
A36: now__::_thesis:_not_i2_-'_1_<_1
assume  i2 -' 1 < 1 ; ::_thesis: contradiction
then  i2 <= 1 by NAT_1:14, NAT_D:36;
then  i2 = 1 by A25, XXREAL_0:1;
then  cell ((Gauge (C,n)),(1 -' 1),j1) meets C by A13, A17, A14, A18, A19, A20, A21, A35, GOBRD13:21;
then  cell ((Gauge (C,n)),0,j1) meets C by XREAL_1:232;
hence  contradiction by A6, A26, JORDAN8:18; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
assume that
A37: [i19,j19] in Indices (Gauge (C,n)) and
A38: [i29,j29] in Indices (Gauge (C,n)) and
A39: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A40: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
A41: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A39, FINSEQ_4:68;
A42: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A40, FINSEQ_4:68;
then A43: j2 = j29 by A20, A21, A38, GOBOARD1:5;
 i2 = i29 by A20, A21, A38, A42, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) ) by A18, A19, A28, A30, A33, A15, A35, A37, A41, A43, A36, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 turns_left (len f) -' 1, Gauge (C,n) by GOBRD13:def_7; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A34; ::_thesis: verum
end;
supposeA44: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_turns_left_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2; ::_thesis: ex j being Element of NAT st
( f1 turns_left (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2 + 1; ::_thesis: ( f1 turns_left (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)
A45: now__::_thesis:_not_j2_+_1_>_len_(Gauge_(C,n))
assume  j2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A46: (len (Gauge (C,n))) + 1 <= j2 + 1 by NAT_1:13;
 j2 + 1 <= (len (Gauge (C,n))) + 1 by A6, A30, XREAL_1:6;
then  j2 + 1 = (len (Gauge (C,n))) + 1 by A46, XXREAL_0:1;
then  cell ((Gauge (C,n)),i1,(len (Gauge (C,n)))) meets C by A13, A17, A14, A18, A19, A20, A21, A44, GOBRD13:23;
hence  contradiction by A23, JORDAN8:15; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
assume that
A47: [i19,j19] in Indices (Gauge (C,n)) and
A48: [i29,j29] in Indices (Gauge (C,n)) and
A49: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A50: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
A51: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A49, FINSEQ_4:68;
A52: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A50, FINSEQ_4:68;
then A53: j2 = j29 by A20, A21, A48, GOBOARD1:5;
 i2 = i29 by A20, A21, A48, A52, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) ) by A6, A18, A19, A25, A32, A24, A15, A44, A47, A51, A53, A45, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 turns_left (len f) -' 1, Gauge (C,n) by GOBRD13:def_7; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A34; ::_thesis: verum
end;
supposeA54: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_turns_left_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2; ::_thesis: ex j being Element of NAT st
( f1 turns_left (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2 -' 1; ::_thesis: ( f1 turns_left (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)
A55: now__::_thesis:_not_j2_-'_1_<_1
assume  j2 -' 1 < 1 ; ::_thesis: contradiction
then  j2 <= 1 by NAT_1:14, NAT_D:36;
then  j2 = 1 by A28, XXREAL_0:1;
then  cell ((Gauge (C,n)),i2,(1 -' 1)) meets C by A13, A17, A14, A18, A19, A20, A21, A54, GOBRD13:25;
then  cell ((Gauge (C,n)),i2,0) meets C by XREAL_1:232;
hence  contradiction by A32, JORDAN8:17; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
assume that
A56: [i19,j19] in Indices (Gauge (C,n)) and
A57: [i29,j29] in Indices (Gauge (C,n)) and
A58: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A59: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
A60: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A58, FINSEQ_4:68;
A61: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A59, FINSEQ_4:68;
then A62: j2 = j29 by A20, A21, A57, GOBOARD1:5;
 i2 = i29 by A20, A21, A57, A61, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) ) by A18, A19, A25, A32, A31, A15, A54, A56, A60, A62, A55, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 turns_left (len f) -' 1, Gauge (C,n) by GOBRD13:def_7; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A34; ::_thesis: verum
end;
supposeA63: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_turns_left_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2 + 1; ::_thesis: ex j being Element of NAT st
( f1 turns_left (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2; ::_thesis: ( f1 turns_left (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)
A64: now__::_thesis:_not_i2_+_1_>_len_(Gauge_(C,n))
assume  i2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A65: (len (Gauge (C,n))) + 1 <= i2 + 1 by NAT_1:13;
 i2 + 1 <= (len (Gauge (C,n))) + 1 by A32, XREAL_1:6;
then  i2 + 1 = (len (Gauge (C,n))) + 1 by A65, XXREAL_0:1;
then  cell ((Gauge (C,n)),(len (Gauge (C,n))),j2) meets C by A13, A17, A14, A18, A19, A20, A21, A63, GOBRD13:27;
hence  contradiction by A6, A30, JORDAN8:16; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
assume that
A66: [i19,j19] in Indices (Gauge (C,n)) and
A67: [i29,j29] in Indices (Gauge (C,n)) and
A68: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A69: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
A70: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A68, FINSEQ_4:68;
A71: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A69, FINSEQ_4:68;
then A72: j2 = j29 by A20, A21, A67, GOBOARD1:5;
 i2 = i29 by A20, A21, A67, A71, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) ) by A18, A19, A28, A30, A27, A15, A63, A66, A70, A72, A64, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 turns_left (len f) -' 1, Gauge (C,n) by GOBRD13:def_7; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A34; ::_thesis: verum
end;
end;
end;
end;
supposeA73: ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ex y being set st S1[k,x,y]
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A22;
supposeA74: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_goes_straight_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2; ::_thesis: ex j being Element of NAT st
( f1 goes_straight (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2 + 1; ::_thesis: ( f1 goes_straight (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)
A75: now__::_thesis:_not_j2_+_1_>_len_(Gauge_(C,n))
assume  j2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A76: (len (Gauge (C,n))) + 1 <= j2 + 1 by NAT_1:13;
 j2 + 1 <= (len (Gauge (C,n))) + 1 by A6, A30, XREAL_1:6;
then  j2 + 1 = (len (Gauge (C,n))) + 1 by A76, XXREAL_0:1;
then  cell ((Gauge (C,n)),(i1 -' 1),(len (Gauge (C,n)))) meets C by A13, A17, A14, A18, A19, A20, A21, A73, A74, GOBRD13:34;
hence  contradiction by A23, JORDAN8:15, NAT_D:44; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) & not ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) & not ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) implies ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
assume that
A77: [i19,j19] in Indices (Gauge (C,n)) and
A78: [i29,j29] in Indices (Gauge (C,n)) and
A79: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A80: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
A81: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A79, FINSEQ_4:68;
A82: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A80, FINSEQ_4:68;
then A83: j2 = j29 by A20, A21, A78, GOBOARD1:5;
 i2 = i29 by A20, A21, A78, A82, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) ) by A6, A18, A19, A25, A32, A24, A15, A74, A77, A81, A83, A75, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 goes_straight (len f) -' 1, Gauge (C,n) by GOBRD13:def_8; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A73; ::_thesis: verum
end;
supposeA84: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_goes_straight_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2 + 1; ::_thesis: ex j being Element of NAT st
( f1 goes_straight (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2; ::_thesis: ( f1 goes_straight (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)
A85: now__::_thesis:_not_i2_+_1_>_len_(Gauge_(C,n))
assume  i2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A86: (len (Gauge (C,n))) + 1 <= i2 + 1 by NAT_1:13;
 i2 + 1 <= (len (Gauge (C,n))) + 1 by A32, XREAL_1:6;
then  i2 + 1 = (len (Gauge (C,n))) + 1 by A86, XXREAL_0:1;
then  cell ((Gauge (C,n)),(len (Gauge (C,n))),j1) meets C by A13, A17, A14, A18, A19, A20, A21, A73, A84, GOBRD13:36;
hence  contradiction by A6, A26, JORDAN8:16; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) & not ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) & not ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) implies ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
assume that
A87: [i19,j19] in Indices (Gauge (C,n)) and
A88: [i29,j29] in Indices (Gauge (C,n)) and
A89: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A90: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
A91: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A89, FINSEQ_4:68;
A92: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A90, FINSEQ_4:68;
then A93: j2 = j29 by A20, A21, A88, GOBOARD1:5;
 i2 = i29 by A20, A21, A88, A92, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) ) by A18, A19, A28, A30, A27, A15, A84, A87, A91, A93, A85, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 goes_straight (len f) -' 1, Gauge (C,n) by GOBRD13:def_8; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A73; ::_thesis: verum
end;
supposeA94: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_goes_straight_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2 -' 1; ::_thesis: ex j being Element of NAT st
( f1 goes_straight (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2; ::_thesis: ( f1 goes_straight (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)
A95: now__::_thesis:_not_i2_-'_1_<_1
assume  i2 -' 1 < 1 ; ::_thesis: contradiction
then  i2 <= 1 by NAT_1:14, NAT_D:36;
then  i2 = 1 by A25, XXREAL_0:1;
then  cell ((Gauge (C,n)),(1 -' 1),(j1 -' 1)) meets C by A13, A17, A14, A18, A19, A20, A21, A73, A94, GOBRD13:38;
then  cell ((Gauge (C,n)),0,(j1 -' 1)) meets C by XREAL_1:232;
hence  contradiction by A6, A26, JORDAN8:18, NAT_D:44; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) & not ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) & not ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) implies ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
assume that
A96: [i19,j19] in Indices (Gauge (C,n)) and
A97: [i29,j29] in Indices (Gauge (C,n)) and
A98: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A99: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
A100: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A98, FINSEQ_4:68;
A101: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A99, FINSEQ_4:68;
then A102: j2 = j29 by A20, A21, A97, GOBOARD1:5;
 i2 = i29 by A20, A21, A97, A101, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) ) by A18, A19, A28, A30, A33, A15, A94, A96, A100, A102, A95, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 goes_straight (len f) -' 1, Gauge (C,n) by GOBRD13:def_8; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A73; ::_thesis: verum
end;
supposeA103: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_goes_straight_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2; ::_thesis: ex j being Element of NAT st
( f1 goes_straight (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2 -' 1; ::_thesis: ( f1 goes_straight (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)
A104: now__::_thesis:_not_j2_-'_1_<_1
assume  j2 -' 1 < 1 ; ::_thesis: contradiction
then  j2 <= 1 by NAT_1:14, NAT_D:36;
then  j2 = 1 by A28, XXREAL_0:1;
then  cell ((Gauge (C,n)),i1,(1 -' 1)) meets C by A13, A17, A14, A18, A19, A20, A21, A73, A103, GOBRD13:40;
then  cell ((Gauge (C,n)),i1,0) meets C by XREAL_1:232;
hence  contradiction by A23, JORDAN8:17; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) & not ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) & not ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) implies ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
assume that
A105: [i19,j19] in Indices (Gauge (C,n)) and
A106: [i29,j29] in Indices (Gauge (C,n)) and
A107: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A108: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
A109: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A107, FINSEQ_4:68;
A110: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A108, FINSEQ_4:68;
then A111: j2 = j29 by A20, A21, A106, GOBOARD1:5;
 i2 = i29 by A20, A21, A106, A110, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) ) by A18, A19, A25, A32, A31, A15, A103, A105, A109, A111, A104, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 goes_straight (len f) -' 1, Gauge (C,n) by GOBRD13:def_8; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A73; ::_thesis: verum
end;
end;
end;
end;
supposeA112: front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ex y being set st S1[k,x,y]
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A22;
supposeA113: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_turns_right_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2 + 1; ::_thesis: ex j being Element of NAT st
( f1 turns_right (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2; ::_thesis: ( f1 turns_right (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)
A114: now__::_thesis:_not_i2_+_1_>_len_(Gauge_(C,n))
assume  i2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A115: (len (Gauge (C,n))) + 1 <= i2 + 1 by NAT_1:13;
 i2 + 1 <= (len (Gauge (C,n))) + 1 by A32, XREAL_1:6;
then  i2 + 1 = (len (Gauge (C,n))) + 1 by A115, XXREAL_0:1;
then  cell ((Gauge (C,n)),(len (Gauge (C,n))),j2) meets C by A13, A17, A14, A18, A19, A20, A21, A112, A113, GOBRD13:35;
hence  contradiction by A6, A30, JORDAN8:16; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
assume that
A116: [i19,j19] in Indices (Gauge (C,n)) and
A117: [i29,j29] in Indices (Gauge (C,n)) and
A118: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A119: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
A120: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A118, FINSEQ_4:68;
A121: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A119, FINSEQ_4:68;
then A122: j2 = j29 by A20, A21, A117, GOBOARD1:5;
 i2 = i29 by A20, A21, A117, A121, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) ) by A18, A19, A28, A30, A27, A15, A113, A116, A120, A122, A114, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 turns_right (len f) -' 1, Gauge (C,n) by GOBRD13:def_6; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A112; ::_thesis: verum
end;
supposeA123: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_turns_right_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2; ::_thesis: ex j being Element of NAT st
( f1 turns_right (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2 -' 1; ::_thesis: ( f1 turns_right (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)
A124: now__::_thesis:_not_j2_-'_1_<_1
assume  j2 -' 1 < 1 ; ::_thesis: contradiction
then  j2 <= 1 by NAT_1:14, NAT_D:36;
then  j2 = 1 by A28, XXREAL_0:1;
then  cell ((Gauge (C,n)),i2,(1 -' 1)) meets C by A13, A17, A14, A18, A19, A20, A21, A112, A123, GOBRD13:37;
then  cell ((Gauge (C,n)),i2,0) meets C by XREAL_1:232;
hence  contradiction by A32, JORDAN8:17; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
assume that
A125: [i19,j19] in Indices (Gauge (C,n)) and
A126: [i29,j29] in Indices (Gauge (C,n)) and
A127: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A128: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
A129: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A127, FINSEQ_4:68;
A130: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A128, FINSEQ_4:68;
then A131: j2 = j29 by A20, A21, A126, GOBOARD1:5;
 i2 = i29 by A20, A21, A126, A130, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) ) by A18, A19, A25, A32, A31, A15, A123, A125, A129, A131, A124, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 turns_right (len f) -' 1, Gauge (C,n) by GOBRD13:def_6; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A112; ::_thesis: verum
end;
supposeA132: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_turns_right_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2; ::_thesis: ex j being Element of NAT st
( f1 turns_right (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2 + 1; ::_thesis: ( f1 turns_right (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)
A133: now__::_thesis:_not_j2_+_1_>_len_(Gauge_(C,n))
assume  j2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A134: (len (Gauge (C,n))) + 1 <= j2 + 1 by NAT_1:13;
 j2 + 1 <= (len (Gauge (C,n))) + 1 by A6, A30, XREAL_1:6;
then  j2 + 1 = (len (Gauge (C,n))) + 1 by A134, XXREAL_0:1;
then  cell ((Gauge (C,n)),(i2 -' 1),(len (Gauge (C,n)))) meets C by A13, A17, A14, A18, A19, A20, A21, A112, A132, GOBRD13:39;
hence  contradiction by A32, JORDAN8:15, NAT_D:44; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
assume that
A135: [i19,j19] in Indices (Gauge (C,n)) and
A136: [i29,j29] in Indices (Gauge (C,n)) and
A137: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A138: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
A139: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A137, FINSEQ_4:68;
A140: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A138, FINSEQ_4:68;
then A141: j2 = j29 by A20, A21, A136, GOBOARD1:5;
 i2 = i29 by A20, A21, A136, A140, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) ) by A6, A18, A19, A25, A32, A24, A15, A132, A135, A139, A141, A133, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 turns_right (len f) -' 1, Gauge (C,n) by GOBRD13:def_6; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A112; ::_thesis: verum
end;
supposeA142: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_
(_f1_turns_right_(len_f)_-'_1,_Gauge_(C,n)_&_f1_=_f_^_<*((Gauge_(C,n))_*_(i,j))*>_)
take i = i2 -' 1; ::_thesis: ex j being Element of NAT st
( f1 turns_right (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
take j = j2; ::_thesis: ( f1 turns_right (len f) -' 1, Gauge (C,n) & f1 = f ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_f1_/._((len_f)_-'_1)_=_(Gauge_(C,n))_*_(i19,j19)_&_f1_/._(((len_f)_-'_1)_+_1)_=_(Gauge_(C,n))_*_(i29,j29)_&_not_(_i19_=_i29_&_j19_+_1_=_j29_&_[(i29_+_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_+_1),j29)_)_&_not_(_i19_+_1_=_i29_&_j19_=_j29_&_[i29,(j29_-'_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_-'_1))_)_&_not_(_i19_=_i29_+_1_&_j19_=_j29_&_[i29,(j29_+_1)]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_(i29,(j29_+_1))_)_holds_
(_i19_=_i29_&_j19_=_j29_+_1_&_[(i29_-'_1),j29]_in_Indices_(Gauge_(C,n))_&_f1_/._(((len_f)_-'_1)_+_2)_=_(Gauge_(C,n))_*_((i29_-'_1),j29)_)
A143: now__::_thesis:_not_i2_-'_1_<_1
assume  i2 -' 1 < 1 ; ::_thesis: contradiction
then  i2 <= 1 by NAT_1:14, NAT_D:36;
then  i2 = 1 by A25, XXREAL_0:1;
then  cell ((Gauge (C,n)),(1 -' 1),(j2 -' 1)) meets C by A13, A17, A14, A18, A19, A20, A21, A112, A142, GOBRD13:41;
then  cell ((Gauge (C,n)),0,(j2 -' 1)) meets C by XREAL_1:232;
hence  contradiction by A6, A30, JORDAN8:18, NAT_D:44; ::_thesis: verum
end;
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) & f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
assume that
A144: [i19,j19] in Indices (Gauge (C,n)) and
A145: [i29,j29] in Indices (Gauge (C,n)) and
A146: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A147: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
A148: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A29, A146, FINSEQ_4:68;
A149: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A16, A147, FINSEQ_4:68;
then A150: j2 = j29 by A20, A21, A145, GOBOARD1:5;
 i2 = i29 by A20, A21, A145, A149, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) ) by A18, A19, A28, A30, A33, A15, A142, A144, A148, A150, A143, FINSEQ_4:67, GOBOARD1:5, MATRIX_1:36; ::_thesis: verum
end;
hence  f1 turns_right (len f) -' 1, Gauge (C,n) by GOBRD13:def_6; ::_thesis: f1 = f ^ <*((Gauge (C,n)) * (i,j))*>
thus  f1 = f ^ <*((Gauge (C,n)) * (i,j))*> ; ::_thesis: verum
end;
hence  S1[k,x,f1] by A10, A12, A13, A112; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
supposeA151: ( not f is_sequence_on Gauge (C,n) or left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ex y being set st S1[k,x,y]
take  f ^ <*((Gauge (C,n)) * (1,1))*> ; ::_thesis: S1[k,x,f ^ <*((Gauge (C,n)) * (1,1))*>]
thus  S1[k,x,f ^ <*((Gauge (C,n)) * (1,1))*>] by A10, A12, A151; ::_thesis: verum
end;
end;
end;
end;
supposeA152: len f <> k ; ::_thesis: ex y being set st S1[k,x,y]
take  {} ; ::_thesis: S1[k,x, {} ]
thus  S1[k,x, {} ] by A10, A152; ::_thesis: verum
end;
end;
end;
end;
supposeA153: ( k > 1 & x is not FinSequence of (TOP-REAL 2) ) ; ::_thesis: ex y being set st S1[k,x,y]
take  {} ; ::_thesis: S1[k,x, {} ]
thus  S1[k,x, {} ] by A153; ::_thesis: verum
end;
end;
end;
consider F being Function such that
A154: dom F = NAT and
A155: F . 0 = {} and
A156: for k being Element of NAT holds S1[k,F . k,F . (k + 1)] from RECDEF_1:sch_1(A7);
defpred S2[ Element of NAT ] means F . $1 is FinSequence of (TOP-REAL 2);
A157: {} = <*> the carrier of (TOP-REAL 2) ;
A158: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A159: F . k is FinSequence of (TOP-REAL 2) ; ::_thesis: S2[k + 1]
percases ( k = 0 or k = 1 or k > 1 ) by NAT_1:25;
suppose k = 0 ; ::_thesis: S2[k + 1]
hence  S2[k + 1] by A156; ::_thesis: verum
end;
suppose k = 1 ; ::_thesis: S2[k + 1]
hence  S2[k + 1] by A156; ::_thesis: verum
end;
supposeA160: k > 1 ; ::_thesis: S2[k + 1]
thus  S2[k + 1] ::_thesis: verum
proof
reconsider f = F . k as FinSequence of (TOP-REAL 2) by A159;
percases ( len f = k or len f <> k ) ;
supposeA161: len f = k ; ::_thesis: S2[k + 1]
thus  S2[k + 1] ::_thesis: verum
proof
percases ( ( f is_sequence_on Gauge (C,n) & left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) or not f is_sequence_on Gauge (C,n) or left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) ;
supposeA162: ( f is_sequence_on Gauge (C,n) & left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: S2[k + 1]
then A163: ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len f) -' 1, Gauge (C,n) & F . (k + 1) = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A156, A160, A161;
A164: ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len f) -' 1, Gauge (C,n) & F . (k + 1) = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A156, A160, A161, A162;
 ( front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len f) -' 1, Gauge (C,n) & F . (k + 1) = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A156, A160, A161, A162;
hence  S2[k + 1] by A164, A163; ::_thesis: verum
end;
supposeA165: ( not f is_sequence_on Gauge (C,n) or left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: S2[k + 1]
 f ^ <*((Gauge (C,n)) * (1,1))*> is FinSequence of (TOP-REAL 2) ;
hence  S2[k + 1] by A156, A160, A161, A165; ::_thesis: verum
end;
end;
end;
end;
suppose len f <> k ; ::_thesis: S2[k + 1]
hence  S2[k + 1] by A156, A157, A160; ::_thesis: verum
end;
end;
end;
end;
end;
end;
A166: S2[ 0 ] by A155, A157;
A167: for k being Element of NAT holds S2[k] from NAT_1:sch_1(A166, A158);
 rng F c= the carrier of (TOP-REAL 2) *
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng F or y in the carrier of (TOP-REAL 2) * )
assume  y in rng F ; ::_thesis: y in the carrier of (TOP-REAL 2) *
then  ex x being set st
( x in dom F & F . x = y ) by FUNCT_1:def_3;
then  y is FinSequence of (TOP-REAL 2) by A154, A167;
hence  y in the carrier of (TOP-REAL 2) * by FINSEQ_1:def_11; ::_thesis: verum
end;
then reconsider F = F as Function of NAT,( the carrier of (TOP-REAL 2) *) by A154, FUNCT_2:def_1, RELSET_1:4;
defpred S3[ Element of NAT ] means  len (F . $1) = $1;
A168: for k being Element of NAT st S3[k] holds
S3[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S3[k] implies S3[k + 1] )
assume A169: len (F . k) = k ; ::_thesis: S3[k + 1]
A170: S1[k,F . k,F . (k + 1)] by A156;
percases ( k = 0 or k = 1 or k > 1 ) by NAT_1:25;
suppose k = 0 ; ::_thesis: S3[k + 1]
hence  S3[k + 1] by A170, FINSEQ_1:39; ::_thesis: verum
end;
suppose k = 1 ; ::_thesis: S3[k + 1]
hence  S3[k + 1] by A170, FINSEQ_1:44; ::_thesis: verum
end;
supposeA171: k > 1 ; ::_thesis: S3[k + 1]
thus  S3[k + 1] ::_thesis: verum
proof
percases ( ( F . k is_sequence_on Gauge (C,n) & left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or not F . k is_sequence_on Gauge (C,n) or left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ;
supposeA172: ( F . k is_sequence_on Gauge (C,n) & left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: S3[k + 1]
then A173: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A156, A169, A171;
A174: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A156, A169, A171, A172;
 ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C implies ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A156, A169, A171, A172;
hence  S3[k + 1] by A169, A174, A173, FINSEQ_2:16; ::_thesis: verum
end;
suppose ( not F . k is_sequence_on Gauge (C,n) or left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: S3[k + 1]
then  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (1,1))*> by A156, A169, A171;
hence  S3[k + 1] by A169, FINSEQ_2:16; ::_thesis: verum
end;
end;
end;
end;
end;
end;
defpred S4[ Element of NAT ] means ( F . $1 is_sequence_on Gauge (C,n) & ( for m being Element of NAT st 1 <= m & m + 1 <= len (F . $1) holds
( right_cell ((F . $1),m,(Gauge (C,n))) misses C & left_cell ((F . $1),m,(Gauge (C,n))) meets C ) ) );
A175: S3[ 0 ] by A155, CARD_1:27;
A176: for k being Element of NAT holds S3[k] from NAT_1:sch_1(A175, A168);
A177: 1 <= X-SpanStart (C,n) by JORDAN1H:49, XXREAL_0:2;
A178: for k being Element of NAT st S4[k] holds
S4[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S4[k] implies S4[k + 1] )
assume that
A179: F . k is_sequence_on Gauge (C,n) and
A180: for m being Element of NAT st 1 <= m & m + 1 <= len (F . k) holds
( right_cell ((F . k),m,(Gauge (C,n))) misses C & left_cell ((F . k),m,(Gauge (C,n))) meets C ) ; ::_thesis: S4[k + 1]
A181: len (F . k) = k by A176;
A182: len (F . (k + 1)) = k + 1 by A176;
percases ( k = 0 or k = 1 or k > 1 ) by NAT_1:25;
supposeA183: k = 0 ; ::_thesis: S4[k + 1]
then A184: F . (k + 1) = <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))))*> by A156;
A185: now__::_thesis:_for_l_being_Element_of_NAT_st_l_in_dom_(F_._(k_+_1))_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_(F_._(k_+_1))_/._l_=_(Gauge_(C,n))_*_(i,j)_)
let l be Element of NAT ; ::_thesis: ( l in dom (F . (k + 1)) implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) ) )
assume A186: l in dom (F . (k + 1)) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) )
then A187: 1 <= l by FINSEQ_3:25;
 l <= 1 by A182, A183, A186, FINSEQ_3:25;
then  l = 1 by A187, XXREAL_0:1;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) ) by A5, A184, FINSEQ_4:16; ::_thesis: verum
end;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_in_dom_(F_._(k_+_1))_&_l_+_1_in_dom_(F_._(k_+_1))_holds_
for_i1,_j1,_i2,_j2_being_Element_of_NAT_st_[i1,j1]_in_Indices_(Gauge_(C,n))_&_[i2,j2]_in_Indices_(Gauge_(C,n))_&_(F_._(k_+_1))_/._l_=_(Gauge_(C,n))_*_(i1,j1)_&_(F_._(k_+_1))_/._(l_+_1)_=_(Gauge_(C,n))_*_(i2,j2)_holds_
(abs_(i1_-_i2))_+_(abs_(j1_-_j2))_=_1
let l be Element of NAT ; ::_thesis: ( l in dom (F . (k + 1)) & l + 1 in dom (F . (k + 1)) implies for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) holds
(abs (i1 - i2)) + (abs (j1 - j2)) = 1 )
assume that
A188: l in dom (F . (k + 1)) and
A189: l + 1 in dom (F . (k + 1)) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) holds
(abs (i1 - i2)) + (abs (j1 - j2)) = 1
A190: 1 <= l by A188, FINSEQ_3:25;
 l <= 1 by A182, A183, A188, FINSEQ_3:25;
then  l = 1 by A190, XXREAL_0:1;
hence  for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) holds
(abs (i1 - i2)) + (abs (j1 - j2)) = 1 by A182, A183, A189, FINSEQ_3:25; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A185, GOBOARD1:def_9; ::_thesis: for m being Element of NAT st 1 <= m & m + 1 <= len (F . (k + 1)) holds
( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 1 <= len (F . (k + 1)) implies ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) )
assume that
A191: 1 <= m and
A192: m + 1 <= len (F . (k + 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 1 <= m + 1 by NAT_1:12;
then  m + 1 = 0 + 1 by A182, A183, A192, XXREAL_0:1;
hence  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) by A191; ::_thesis: verum
end;
supposeA193: k = 1 ; ::_thesis: S4[k + 1]
A194: (X-SpanStart (C,n)) -' 1 < X-SpanStart (C,n) by A177, JORDAN5B:1;
A195: X-SpanStart (C,n) <= (X-SpanStart (C,n)) + 1 by NAT_1:11;
A196: [(X-SpanStart (C,n)),(Y-SpanStart (C,n))] in Indices (Gauge (C,n)) by A1, JORDAN11:8;
A197: F . (k + 1) = <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n)))),((Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))))*> by A156, A193;
then A198: (F . (k + 1)) /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) by FINSEQ_4:17;
A199: [((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))] in Indices (Gauge (C,n)) by A1, JORDAN11:9;
A200: (F . (k + 1)) /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) by A197, FINSEQ_4:17;
A201: now__::_thesis:_for_l_being_Element_of_NAT_st_l_in_dom_(F_._(k_+_1))_&_l_+_1_in_dom_(F_._(k_+_1))_holds_
for_i1,_j1,_i2,_j2_being_Element_of_NAT_st_[i1,j1]_in_Indices_(Gauge_(C,n))_&_[i2,j2]_in_Indices_(Gauge_(C,n))_&_(F_._(k_+_1))_/._l_=_(Gauge_(C,n))_*_(i1,j1)_&_(F_._(k_+_1))_/._(l_+_1)_=_(Gauge_(C,n))_*_(i2,j2)_holds_
(abs_(i1_-_i2))_+_(abs_(j1_-_j2))_=_1
let l be Element of NAT ; ::_thesis: ( l in dom (F . (k + 1)) & l + 1 in dom (F . (k + 1)) implies for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) holds
(abs (i1 - i2)) + (abs (j1 - j2)) = 1 )
assume that
A202: l in dom (F . (k + 1)) and
A203: l + 1 in dom (F . (k + 1)) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) holds
(abs (i1 - i2)) + (abs (j1 - j2)) = 1
let i1, j1, i2, j2 be Element of NAT ; ::_thesis: ( [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) implies (abs (i1 - i2)) + (abs (j1 - j2)) = 1 )
assume that
A204: [i1,j1] in Indices (Gauge (C,n)) and
A205: [i2,j2] in Indices (Gauge (C,n)) and
A206: (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) and
A207: (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) ; ::_thesis: (abs (i1 - i2)) + (abs (j1 - j2)) = 1
 l <= 2 by A182, A193, A202, FINSEQ_3:25;
then A208: ( l = 0 or l = 1 or l = 2 ) by NAT_1:26;
then A209: i2 = (X-SpanStart (C,n)) -' 1 by A182, A193, A200, A199, A202, A203, A205, A207, FINSEQ_3:25, GOBOARD1:5;
A210: j1 = Y-SpanStart (C,n) by A198, A200, A196, A199, A202, A208, A204, A206, FINSEQ_3:25, GOBOARD1:5;
 j2 = Y-SpanStart (C,n) by A182, A193, A198, A200, A196, A199, A203, A208, A205, A207, FINSEQ_3:25, GOBOARD1:5;
then A211: abs (j1 - j2) = 0 by A210, ABSVALUE:def_1;
 i1 = X-SpanStart (C,n) by A182, A193, A198, A196, A202, A203, A208, A204, A206, FINSEQ_3:25, GOBOARD1:5;
then  i2 + 1 = i1 by A3, A209, NAT_D:43, NAT_D:55;
hence  (abs (i1 - i2)) + (abs (j1 - j2)) = 1 by A211, ABSVALUE:def_1; ::_thesis: verum
end;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_in_dom_(F_._(k_+_1))_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_(F_._(k_+_1))_/._l_=_(Gauge_(C,n))_*_(i,j)_)
let l be Element of NAT ; ::_thesis: ( l in dom (F . (k + 1)) implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) ) )
assume A212: l in dom (F . (k + 1)) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) )
then  l <= 2 by A182, A193, FINSEQ_3:25;
then  ( l = 0 or l = 1 or l = 2 ) by NAT_1:26;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) ) by A198, A200, A196, A199, A212, FINSEQ_3:25; ::_thesis: verum
end;
hence A213: F . (k + 1) is_sequence_on Gauge (C,n) by A201, GOBOARD1:def_9; ::_thesis: for m being Element of NAT st 1 <= m & m + 1 <= len (F . (k + 1)) holds
( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 1 <= len (F . (k + 1)) implies ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) )
assume that
A214: 1 <= m and
A215: m + 1 <= len (F . (k + 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 1 + 1 <= m + 1 by A214, XREAL_1:6;
then A216: m + 1 = 1 + 1 by A182, A193, A215, XXREAL_0:1;
then  right_cell ((F . (k + 1)),m,(Gauge (C,n))) = cell ((Gauge (C,n)),((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) by A198, A200, A196, A199, A213, A215, A194, A195, GOBRD13:def_2;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A1, JORDAN11:11; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 left_cell ((F . (k + 1)),m,(Gauge (C,n))) = cell ((Gauge (C,n)),((X-SpanStart (C,n)) -' 1),((Y-SpanStart (C,n)) -' 1)) by A198, A200, A196, A199, A213, A215, A216, A194, A195, GOBRD13:def_3;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A1, JORDAN11:10; ::_thesis: verum
end;
supposeA217: k > 1 ; ::_thesis: S4[k + 1]
then A218: len (F . k) in dom (F . k) by A181, FINSEQ_3:25;
A219: ((len (F . k)) -' 1) + 1 = len (F . k) by A181, A217, XREAL_1:235;
then A220: ((len (F . k)) -' 1) + (1 + 1) = (len (F . k)) + 1 ;
A221: 1 <= (len (F . k)) -' 1 by A181, A217, NAT_D:49;
then consider i1, j1, i2, j2 being Element of NAT  such that
A222: [i1,j1] in Indices (Gauge (C,n)) and
A223: (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) and
A224: [i2,j2] in Indices (Gauge (C,n)) and
A225: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) and
 ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A179, A219, JORDAN8:3;
A226: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, JORDAN1H:58;
 (i1 + 1) + 1 = i1 + 2 ;
then A227: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, JORDAN1H:57;
 (j1 + 1) + 1 = j1 + 2 ;
then A228: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [i1,(j2 + 1)] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, JORDAN1H:56;
A229: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, JORDAN1H:63;
 (len (F . k)) -' 1 <= len (F . k) by NAT_D:35;
then A230: (len (F . k)) -' 1 in dom (F . k) by A221, FINSEQ_3:25;
A231: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, JORDAN1H:62;
A232: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, JORDAN1H:61;
A233: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, JORDAN1H:60;
A234: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, JORDAN1H:59;
A235: 1 <= j2 by A224, MATRIX_1:38;
A236: left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C by A180, A221, A219;
then A237: ( i1 = i2 & j1 + 1 = j2 implies [(i1 -' 1),(j1 + 1)] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, JORDAN1H:52;
A238: ( i1 = i2 & j1 = j2 + 1 implies [(i1 + 1),j2] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, A236, JORDAN1H:55;
A239: ( i1 = i2 + 1 & j1 = j2 implies [i2,(j1 -' 1)] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, A236, JORDAN1H:54;
A240: ( i1 + 1 = i2 & j1 = j2 implies [(i1 + 1),(j1 + 1)] in Indices (Gauge (C,n)) ) by A179, A181, A217, A222, A223, A224, A225, A236, JORDAN1H:53;
A241: 1 <= i2 by A224, MATRIX_1:38;
thus A242: F . (k + 1) is_sequence_on Gauge (C,n) ::_thesis: for m being Element of NAT st 1 <= m & m + 1 <= len (F . (k + 1)) holds
( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
proof
percases ( ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) or ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ;
suppose ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
then consider i, j being Element of NAT  such that
A243: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len (F . k)) -' 1, Gauge (C,n) and
A244: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A156, A179, A181, A217, A236;
set f = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>;
A245: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by FINSEQ_4:67;
A246: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A225, A218, FINSEQ_4:68;
A247: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A223, A230, FINSEQ_4:68;
thus  F . (k + 1) is_sequence_on Gauge (C,n) ::_thesis: verum
proof
percases ( ( i1 = i2 & j1 + 1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) or ( i1 + 1 = i2 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 = i2 + 1 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) or ( i1 = i2 & j1 = j2 + 1 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) ) by A219, A222, A224, A220, A243, A247, A246, GOBRD13:def_7;
supposethat A248: ( i1 = i2 & j1 + 1 = j2 ) and
A249: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_-'_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A250: [i19,j19] in Indices (Gauge (C,n)) and
A251: [i29,j29] in Indices (Gauge (C,n)) and
A252: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A253: (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A254: i2 -' 1 = i29 by A237, A248, A251, A253, GOBOARD1:5;
 i2 = i19 by A224, A225, A250, A252, GOBOARD1:5;
then  i19 - i29 = i2 - (i2 - 1) by A241, A254, XREAL_1:233;
then A255: abs (i19 - i29) = 1 by ABSVALUE:def_1;
A256: j2 = j29 by A237, A248, A251, A253, GOBOARD1:5;
 j2 = j19 by A224, A225, A250, A252, GOBOARD1:5;
then  abs (j29 - j19) = 0 by A256, ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A255, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A237, A244, A245, A248, A249, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A257: ( i1 + 1 = i2 & j1 = j2 ) and
A258: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_+_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A259: [i19,j19] in Indices (Gauge (C,n)) and
A260: [i29,j29] in Indices (Gauge (C,n)) and
A261: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A262: (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A263: i2 = i29 by A240, A257, A260, A262, GOBOARD1:5;
 i2 = i19 by A224, A225, A259, A261, GOBOARD1:5;
then A264: abs (i29 - i19) = 0 by A263, ABSVALUE:def_1;
A265: j2 + 1 = j29 by A240, A257, A260, A262, GOBOARD1:5;
 j2 = j19 by A224, A225, A259, A261, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A265, A264, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A240, A244, A245, A257, A258, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A266: ( i1 = i2 + 1 & j1 = j2 ) and
A267: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_-'_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A268: [i19,j19] in Indices (Gauge (C,n)) and
A269: [i29,j29] in Indices (Gauge (C,n)) and
A270: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A271: (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A272: j2 -' 1 = j29 by A239, A266, A269, A271, GOBOARD1:5;
 j2 = j19 by A224, A225, A268, A270, GOBOARD1:5;
then  j19 - j29 = j2 - (j2 - 1) by A235, A272, XREAL_1:233;
then A273: abs (j19 - j29) = 1 by ABSVALUE:def_1;
A274: i2 = i29 by A239, A266, A269, A271, GOBOARD1:5;
 i2 = i19 by A224, A225, A268, A270, GOBOARD1:5;
then  abs (i29 - i19) = 0 by A274, ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A273, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A239, A244, A245, A266, A267, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A275: ( i1 = i2 & j1 = j2 + 1 ) and
A276: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_+_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A277: [i19,j19] in Indices (Gauge (C,n)) and
A278: [i29,j29] in Indices (Gauge (C,n)) and
A279: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A280: (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A281: j2 = j29 by A238, A275, A278, A280, GOBOARD1:5;
 j2 = j19 by A224, A225, A277, A279, GOBOARD1:5;
then A282: abs (j29 - j19) = 0 by A281, ABSVALUE:def_1;
A283: i2 + 1 = i29 by A238, A275, A278, A280, GOBOARD1:5;
 i2 = i19 by A224, A225, A277, A279, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A283, A282, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A238, A244, A245, A275, A276, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
end;
end;
end;
supposeA284: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
then consider i, j being Element of NAT  such that
A285: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len (F . k)) -' 1, Gauge (C,n) and
A286: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A156, A179, A181, A217, A236;
set f = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>;
A287: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by FINSEQ_4:67;
A288: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A225, A218, FINSEQ_4:68;
A289: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A223, A230, FINSEQ_4:68;
thus  F . (k + 1) is_sequence_on Gauge (C,n) ::_thesis: verum
proof
percases ( ( i1 = i2 & j1 + 1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 + 1 = i2 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) or ( i1 = i2 + 1 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) or ( i1 = i2 & j1 = j2 + 1 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) ) by A219, A222, A224, A220, A285, A289, A288, GOBRD13:def_8;
supposethat A290: ( i1 = i2 & j1 + 1 = j2 ) and
A291: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_+_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A292: [i19,j19] in Indices (Gauge (C,n)) and
A293: [i29,j29] in Indices (Gauge (C,n)) and
A294: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A295: (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A296: i2 = i19 by A224, A225, A292, A294, GOBOARD1:5;
 i2 = i29 by A228, A284, A290, A293, A295, GOBOARD1:5;
then A297: abs (i29 - i19) = 0 by A296, ABSVALUE:def_1;
A298: j2 = j19 by A224, A225, A292, A294, GOBOARD1:5;
 j2 + 1 = j29 by A228, A284, A290, A293, A295, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A298, A297, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A228, A284, A286, A287, A290, A291, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A299: ( i1 + 1 = i2 & j1 = j2 ) and
A300: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_+_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A301: [i19,j19] in Indices (Gauge (C,n)) and
A302: [i29,j29] in Indices (Gauge (C,n)) and
A303: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A304: (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A305: j2 = j19 by A224, A225, A301, A303, GOBOARD1:5;
 j2 = j29 by A227, A284, A299, A302, A304, GOBOARD1:5;
then A306: abs (j29 - j19) = 0 by A305, ABSVALUE:def_1;
A307: i2 = i19 by A224, A225, A301, A303, GOBOARD1:5;
 i2 + 1 = i29 by A227, A284, A299, A302, A304, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A307, A306, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A227, A284, A286, A287, A299, A300, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A308: ( i1 = i2 + 1 & j1 = j2 ) and
A309: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_-'_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A310: [i19,j19] in Indices (Gauge (C,n)) and
A311: [i29,j29] in Indices (Gauge (C,n)) and
A312: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A313: (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A314: i2 = i19 by A224, A225, A310, A312, GOBOARD1:5;
 i2 -' 1 = i29 by A226, A284, A308, A311, A313, GOBOARD1:5;
then  i19 - i29 = i2 - (i2 - 1) by A241, A314, XREAL_1:233;
then A315: abs (i19 - i29) = 1 by ABSVALUE:def_1;
A316: j2 = j19 by A224, A225, A310, A312, GOBOARD1:5;
 j2 = j29 by A226, A284, A308, A311, A313, GOBOARD1:5;
then  abs (j29 - j19) = 0 by A316, ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A315, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A226, A284, A286, A287, A308, A309, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A317: ( i1 = i2 & j1 = j2 + 1 ) and
A318: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_-'_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A319: [i19,j19] in Indices (Gauge (C,n)) and
A320: [i29,j29] in Indices (Gauge (C,n)) and
A321: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A322: (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A323: j2 = j19 by A224, A225, A319, A321, GOBOARD1:5;
 j2 -' 1 = j29 by A234, A284, A317, A320, A322, GOBOARD1:5;
then  j19 - j29 = j2 - (j2 - 1) by A235, A323, XREAL_1:233;
then A324: abs (j19 - j29) = 1 by ABSVALUE:def_1;
A325: i2 = i19 by A224, A225, A319, A321, GOBOARD1:5;
 i2 = i29 by A234, A284, A317, A320, A322, GOBOARD1:5;
then  abs (i29 - i19) = 0 by A325, ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A324, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A234, A284, A286, A287, A317, A318, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
end;
end;
end;
supposeA326: front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
then consider i, j being Element of NAT  such that
A327: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len (F . k)) -' 1, Gauge (C,n) and
A328: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A156, A179, A181, A217, A236;
set f = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>;
A329: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by FINSEQ_4:67;
A330: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A225, A218, FINSEQ_4:68;
A331: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A223, A230, FINSEQ_4:68;
thus  F . (k + 1) is_sequence_on Gauge (C,n) ::_thesis: verum
proof
percases ( ( i1 = i2 & j1 + 1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) or ( i1 + 1 = i2 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) or ( i1 = i2 + 1 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 = i2 & j1 = j2 + 1 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) ) by A219, A222, A224, A220, A327, A331, A330, GOBRD13:def_6;
supposethat A332: ( i1 = i2 & j1 + 1 = j2 ) and
A333: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_+_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A334: [i19,j19] in Indices (Gauge (C,n)) and
A335: [i29,j29] in Indices (Gauge (C,n)) and
A336: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A337: (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A338: j2 = j19 by A224, A225, A334, A336, GOBOARD1:5;
 j2 = j29 by A233, A326, A332, A335, A337, GOBOARD1:5;
then A339: abs (j29 - j19) = 0 by A338, ABSVALUE:def_1;
A340: i2 = i19 by A224, A225, A334, A336, GOBOARD1:5;
 i2 + 1 = i29 by A233, A326, A332, A335, A337, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A340, A339, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A233, A326, A328, A329, A332, A333, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A341: ( i1 + 1 = i2 & j1 = j2 ) and
A342: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_-'_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A343: [i19,j19] in Indices (Gauge (C,n)) and
A344: [i29,j29] in Indices (Gauge (C,n)) and
A345: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A346: (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A347: j2 = j19 by A224, A225, A343, A345, GOBOARD1:5;
 j2 -' 1 = j29 by A232, A326, A341, A344, A346, GOBOARD1:5;
then  j19 - j29 = j2 - (j2 - 1) by A235, A347, XREAL_1:233;
then A348: abs (j19 - j29) = 1 by ABSVALUE:def_1;
A349: i2 = i19 by A224, A225, A343, A345, GOBOARD1:5;
 i2 = i29 by A232, A326, A341, A344, A346, GOBOARD1:5;
then  abs (i29 - i19) = 0 by A349, ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A348, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A232, A326, A328, A329, A341, A342, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A350: ( i1 = i2 + 1 & j1 = j2 ) and
A351: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_+_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A352: [i19,j19] in Indices (Gauge (C,n)) and
A353: [i29,j29] in Indices (Gauge (C,n)) and
A354: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A355: (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A356: i2 = i19 by A224, A225, A352, A354, GOBOARD1:5;
 i2 = i29 by A231, A326, A350, A353, A355, GOBOARD1:5;
then A357: abs (i29 - i19) = 0 by A356, ABSVALUE:def_1;
A358: j2 = j19 by A224, A225, A352, A354, GOBOARD1:5;
 j2 + 1 = j29 by A231, A326, A350, A353, A355, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A358, A357, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A231, A326, A328, A329, A350, A351, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A359: ( i1 = i2 & j1 = j2 + 1 ) and
A360: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_-'_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume that
A361: [i19,j19] in Indices (Gauge (C,n)) and
A362: [i29,j29] in Indices (Gauge (C,n)) and
A363: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) and
A364: (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
A365: i2 = i19 by A224, A225, A361, A363, GOBOARD1:5;
 i2 -' 1 = i29 by A229, A326, A359, A362, A364, GOBOARD1:5;
then  i19 - i29 = i2 - (i2 - 1) by A241, A365, XREAL_1:233;
then A366: abs (i19 - i29) = 1 by ABSVALUE:def_1;
A367: j2 = j19 by A224, A225, A361, A363, GOBOARD1:5;
 j2 = j29 by A229, A326, A359, A362, A364, GOBOARD1:5;
then  abs (j29 - j19) = 0 by A367, ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A366, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A179, A181, A217, A229, A326, A328, A329, A359, A360, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
end;
end;
end;
end;
end;
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 1 <= len (F . (k + 1)) implies ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) )
assume that
A368: 1 <= m and
A369: m + 1 <= len (F . (k + 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
A370: right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C by A180, A221, A219;
now__::_thesis:_(_right_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( m + 1 = len (F . (k + 1)) or m + 1 <> len (F . (k + 1)) ) ;
supposeA371: m + 1 = len (F . (k + 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
A372: (j2 -' 1) + 1 = j2 by A235, XREAL_1:235;
A373: (i2 -' 1) + 1 = i2 by A241, XREAL_1:235;
thus  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ::_thesis: verum
proof
percases ( ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) or ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ;
supposeA374: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then A375: ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A156, A179, A181, A217, A236;
then A376: (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A225, A218, FINSEQ_4:68;
A377: (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A223, A230, A375, FINSEQ_4:68;
now__::_thesis:_(_right_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( ( i1 = i2 & j1 + 1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) or ( i1 + 1 = i2 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 = i2 + 1 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) or ( i1 = i2 & j1 = j2 + 1 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) ) by A219, A222, A224, A220, A375, A377, A376, GOBRD13:def_7;
supposethat A378: ( i1 = i2 & j1 + 1 = j2 ) and
A379: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j2) by A179, A221, A219, A222, A223, A224, A225, A378, GOBRD13:34;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A237, A242, A368, A371, A373, A374, A376, A378, A379, GOBRD13:26; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
A380: j2 -' 1 = j1 by A378, NAT_D:34;
 cell ((Gauge (C,n)),(i1 -' 1),j1) meets C by A179, A221, A219, A222, A223, A224, A225, A236, A378, GOBRD13:21;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A237, A242, A368, A371, A373, A376, A378, A379, A380, GOBRD13:25; ::_thesis: verum
end;
supposethat A381: ( i1 + 1 = i2 & j1 = j2 ) and
A382: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) by A179, A221, A219, A222, A223, A224, A225, A381, GOBRD13:36;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A240, A242, A368, A371, A374, A376, A381, A382, GOBRD13:22; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
A383: (i1 + 1) -' 1 = i1 by NAT_D:34;
 cell ((Gauge (C,n)),i1,j1) meets C by A179, A221, A219, A222, A223, A224, A225, A236, A381, GOBRD13:23;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A240, A242, A368, A371, A376, A381, A382, A383, GOBRD13:21; ::_thesis: verum
end;
supposethat A384: ( i1 = i2 + 1 & j1 = j2 ) and
A385: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A179, A221, A219, A222, A223, A224, A225, A384, GOBRD13:38;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A239, A242, A368, A371, A372, A374, A376, A384, A385, GOBRD13:28; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 cell ((Gauge (C,n)),i2,(j2 -' 1)) meets C by A179, A221, A219, A222, A223, A224, A225, A236, A384, GOBRD13:25;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A239, A242, A368, A371, A372, A376, A384, A385, GOBRD13:27; ::_thesis: verum
end;
supposethat A386: ( i1 = i2 & j1 = j2 + 1 ) and
A387: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A179, A221, A219, A222, A223, A224, A225, A386, GOBRD13:40;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A238, A242, A368, A371, A374, A376, A386, A387, GOBRD13:24; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 cell ((Gauge (C,n)),i2,j2) meets C by A179, A221, A219, A222, A223, A224, A225, A236, A386, GOBRD13:27;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A238, A242, A368, A371, A376, A386, A387, GOBRD13:23; ::_thesis: verum
end;
end;
end;
hence  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
supposeA388: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then A389: ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A156, A179, A181, A217, A236;
then A390: (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A225, A218, FINSEQ_4:68;
A391: (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A223, A230, A389, FINSEQ_4:68;
now__::_thesis:_(_right_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( ( i1 = i2 & j1 + 1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 + 1 = i2 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) or ( i1 = i2 + 1 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) or ( i1 = i2 & j1 = j2 + 1 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) ) by A219, A222, A224, A220, A389, A391, A390, GOBRD13:def_8;
supposethat A392: ( i1 = i2 & j1 + 1 = j2 ) and
A393: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j2) by A179, A221, A219, A222, A223, A224, A225, A392, GOBRD13:35;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A228, A242, A368, A371, A388, A390, A392, A393, GOBRD13:22; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j2) by A179, A221, A219, A222, A223, A224, A225, A392, GOBRD13:34;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A228, A242, A368, A371, A388, A390, A392, A393, GOBRD13:21; ::_thesis: verum
end;
supposethat A394: ( i1 + 1 = i2 & j1 = j2 ) and
A395: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A179, A221, A219, A222, A223, A224, A225, A394, GOBRD13:37;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A227, A242, A368, A371, A388, A390, A394, A395, GOBRD13:24; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) by A179, A221, A219, A222, A223, A224, A225, A394, GOBRD13:36;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A227, A242, A368, A371, A388, A390, A394, A395, GOBRD13:23; ::_thesis: verum
end;
supposethat A396: ( i1 = i2 + 1 & j1 = j2 ) and
A397: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),j2) by A179, A221, A219, A222, A223, A224, A225, A396, GOBRD13:39;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A226, A242, A368, A371, A373, A388, A390, A396, A397, GOBRD13:26; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A179, A221, A219, A222, A223, A224, A225, A396, GOBRD13:38;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A226, A242, A368, A371, A373, A388, A390, A396, A397, GOBRD13:25; ::_thesis: verum
end;
supposethat A398: ( i1 = i2 & j1 = j2 + 1 ) and
A399: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A179, A221, A219, A222, A223, A224, A225, A398, GOBRD13:41;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A234, A242, A368, A371, A372, A388, A390, A398, A399, GOBRD13:28; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A179, A221, A219, A222, A223, A224, A225, A398, GOBRD13:40;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A234, A242, A368, A371, A372, A388, A390, A398, A399, GOBRD13:27; ::_thesis: verum
end;
end;
end;
hence  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
supposeA400: front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then A401: ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A156, A179, A181, A217, A236;
then A402: (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A225, A218, FINSEQ_4:68;
A403: (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A223, A230, A401, FINSEQ_4:68;
now__::_thesis:_(_right_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( ( i1 = i2 & j1 + 1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) or ( i1 + 1 = i2 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) or ( i1 = i2 + 1 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 = i2 & j1 = j2 + 1 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) ) by A219, A222, A224, A220, A401, A403, A402, GOBRD13:def_6;
supposethat A404: ( i1 = i2 & j1 + 1 = j2 ) and
A405: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
A406: j2 -' 1 = j1 by A404, NAT_D:34;
 cell ((Gauge (C,n)),i1,j1) misses C by A179, A221, A219, A222, A223, A224, A225, A370, A404, GOBRD13:22;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A233, A242, A368, A371, A400, A402, A404, A405, A406, GOBRD13:24; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) by A179, A221, A219, A222, A223, A224, A225, A404, GOBRD13:35;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A233, A242, A368, A371, A400, A402, A404, A405, GOBRD13:23; ::_thesis: verum
end;
supposethat A407: ( i1 + 1 = i2 & j1 = j2 ) and
A408: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
A409: i2 -' 1 = i1 by A407, NAT_D:34;
 cell ((Gauge (C,n)),i1,(j1 -' 1)) misses C by A179, A221, A219, A222, A223, A224, A225, A370, A407, GOBRD13:24;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A232, A242, A368, A371, A372, A400, A402, A407, A408, A409, GOBRD13:28; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A179, A221, A219, A222, A223, A224, A225, A407, GOBRD13:37;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A232, A242, A368, A371, A372, A400, A402, A407, A408, GOBRD13:27; ::_thesis: verum
end;
supposethat A410: ( i1 = i2 + 1 & j1 = j2 ) and
A411: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 cell ((Gauge (C,n)),i2,j2) misses C by A179, A221, A219, A222, A223, A224, A225, A370, A410, GOBRD13:26;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A231, A242, A368, A371, A400, A402, A410, A411, GOBRD13:22; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),j2) by A179, A221, A219, A222, A223, A224, A225, A410, GOBRD13:39;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A231, A242, A368, A371, A400, A402, A410, A411, GOBRD13:21; ::_thesis: verum
end;
supposethat A412: ( i1 = i2 & j1 = j2 + 1 ) and
A413: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 cell ((Gauge (C,n)),(i2 -' 1),j2) misses C by A179, A221, A219, A222, A223, A224, A225, A370, A412, GOBRD13:28;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A181, A182, A224, A229, A242, A368, A371, A373, A400, A402, A412, A413, GOBRD13:26; ::_thesis: left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A179, A221, A219, A222, A223, A224, A225, A412, GOBRD13:41;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A181, A182, A224, A229, A242, A368, A371, A373, A400, A402, A412, A413, GOBRD13:25; ::_thesis: verum
end;
end;
end;
hence  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
end;
end;
end;
suppose m + 1 <> len (F . (k + 1)) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then  m + 1 < len (F . (k + 1)) by A369, XXREAL_0:1;
then A414: m + 1 <= len (F . k) by A181, A182, NAT_1:13;
then consider i1, j1, i2, j2 being Element of NAT  such that
A415: [i1,j1] in Indices (Gauge (C,n)) and
A416: (F . k) /. m = (Gauge (C,n)) * (i1,j1) and
A417: [i2,j2] in Indices (Gauge (C,n)) and
A418: (F . k) /. (m + 1) = (Gauge (C,n)) * (i2,j2) and
A419: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A179, A368, JORDAN8:3;
A420: right_cell ((F . k),m,(Gauge (C,n))) misses C by A180, A368, A414;
A421: now__::_thesis:_ex_i,_j_being_Element_of_NAT_st_F_._(k_+_1)_=_(F_._k)_^_<*((Gauge_(C,n))_*_(i,j))*>
percases ( ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) or ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ;
suppose ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ex i, j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
then consider i, j being Element of NAT  such that
 (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len (F . k)) -' 1, Gauge (C,n) and
A422: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A156, A179, A181, A217, A236;
take i = i; ::_thesis: ex j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
take j = j; ::_thesis: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
thus  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A422; ::_thesis: verum
end;
suppose ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ex i, j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
then consider i, j being Element of NAT  such that
 (F . k) ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len (F . k)) -' 1, Gauge (C,n) and
A423: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A156, A179, A181, A217, A236;
take i = i; ::_thesis: ex j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
take j = j; ::_thesis: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
thus  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A423; ::_thesis: verum
end;
suppose front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ex i, j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
then consider i, j being Element of NAT  such that
 (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len (F . k)) -' 1, Gauge (C,n) and
A424: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A156, A179, A181, A217, A236;
take i = i; ::_thesis: ex j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
take j = j; ::_thesis: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
thus  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A424; ::_thesis: verum
end;
end;
end;
 1 <= m + 1 by NAT_1:12;
then  m + 1 in dom (F . k) by A414, FINSEQ_3:25;
then A425: (F . (k + 1)) /. (m + 1) = (Gauge (C,n)) * (i2,j2) by A418, A421, FINSEQ_4:68;
A426: left_cell ((F . k),m,(Gauge (C,n))) meets C by A180, A368, A414;
 m <= len (F . k) by A414, NAT_1:13;
then  m in dom (F . k) by A368, FINSEQ_3:25;
then A427: (F . (k + 1)) /. m = (Gauge (C,n)) * (i1,j1) by A416, A421, FINSEQ_4:68;
now__::_thesis:_(_right_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A419;
supposeA428: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then A429: right_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j1) by A179, A368, A414, A415, A416, A417, A418, GOBRD13:22;
 left_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j1) by A179, A368, A414, A415, A416, A417, A418, A428, GOBRD13:21;
hence  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) by A242, A368, A369, A415, A417, A420, A426, A427, A425, A428, A429, GOBRD13:21, GOBRD13:22; ::_thesis: verum
end;
supposeA430: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then A431: right_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,(j1 -' 1)) by A179, A368, A414, A415, A416, A417, A418, GOBRD13:24;
 left_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j1) by A179, A368, A414, A415, A416, A417, A418, A430, GOBRD13:23;
hence  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) by A242, A368, A369, A415, A417, A420, A426, A427, A425, A430, A431, GOBRD13:23, GOBRD13:24; ::_thesis: verum
end;
supposeA432: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then A433: right_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) by A179, A368, A414, A415, A416, A417, A418, GOBRD13:26;
 left_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A179, A368, A414, A415, A416, A417, A418, A432, GOBRD13:25;
hence  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) by A242, A368, A369, A415, A417, A420, A426, A427, A425, A432, A433, GOBRD13:25, GOBRD13:26; ::_thesis: verum
end;
supposeA434: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then A435: right_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j2) by A179, A368, A414, A415, A416, A417, A418, GOBRD13:28;
 left_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) by A179, A368, A414, A415, A416, A417, A418, A434, GOBRD13:27;
hence  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) by A242, A368, A369, A415, A417, A420, A426, A427, A425, A434, A435, GOBRD13:27, GOBRD13:28; ::_thesis: verum
end;
end;
end;
hence  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
end;
end;
hence  ( right_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
end;
end;
A436: S4[ 0 ]
proof
A437: for n being Element of NAT st n in dom (F . 0) & n + 1 in dom (F . 0) holds
for m, k, i, j being Element of NAT st [m,k] in Indices (Gauge (C,n)) & [i,j] in Indices (Gauge (C,n)) & (F . 0) /. n = (Gauge (C,n)) * (m,k) & (F . 0) /. (n + 1) = (Gauge (C,n)) * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1 by A155;
 for n being Element of NAT st n in dom (F . 0) holds
ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . 0) /. n = (Gauge (C,n)) * (i,j) ) by A155;
hence  F . 0 is_sequence_on Gauge (C,n) by A437, GOBOARD1:def_9; ::_thesis: for m being Element of NAT st 1 <= m & m + 1 <= len (F . 0) holds
( right_cell ((F . 0),m,(Gauge (C,n))) misses C & left_cell ((F . 0),m,(Gauge (C,n))) meets C )
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 1 <= len (F . 0) implies ( right_cell ((F . 0),m,(Gauge (C,n))) misses C & left_cell ((F . 0),m,(Gauge (C,n))) meets C ) )
assume that
 1 <= m and
A438: m + 1 <= len (F . 0) ; ::_thesis: ( right_cell ((F . 0),m,(Gauge (C,n))) misses C & left_cell ((F . 0),m,(Gauge (C,n))) meets C )
thus  ( right_cell ((F . 0),m,(Gauge (C,n))) misses C & left_cell ((F . 0),m,(Gauge (C,n))) meets C ) by A155, A438, CARD_1:27; ::_thesis: verum
end;
A439: for k being Element of NAT holds S4[k] from NAT_1:sch_1(A436, A178);
A440: for k, i1, i2, j1, j2 being Element of NAT st k > 1 & [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C holds
( F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) )
proof
let k, i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( k > 1 & [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C implies ( F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) ) )
assume that
A441: k > 1 and
A442: [i1,j1] in Indices (Gauge (C,n)) and
A443: (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) and
A444: [i2,j2] in Indices (Gauge (C,n)) and
A445: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ; ::_thesis: ( not front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C or not front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C or ( F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) ) )
A446: len (F . k) = k by A176;
then A447: 1 <= (len (F . k)) -' 1 by A441, NAT_D:49;
 (len (F . k)) -' 1 <= len (F . k) by NAT_D:35;
then A448: (len (F . k)) -' 1 in dom (F . k) by A447, FINSEQ_3:25;
A449: i1 + 1 > i1 by NAT_1:13;
A450: F . k is_sequence_on Gauge (C,n) by A439;
A451: j1 + 1 > j1 by NAT_1:13;
A452: len (F . k) in dom (F . k) by A441, A446, FINSEQ_3:25;
A453: i2 + 1 > i2 by NAT_1:13;
A454: j2 + 1 > j2 by NAT_1:13;
A455: ((len (F . k)) -' 1) + 1 = len (F . k) by A441, A446, XREAL_1:235;
then A456: ((len (F . k)) -' 1) + (1 + 1) = (len (F . k)) + 1 ;
A457: left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C by A439, A447, A455;
hereby  ::_thesis: verum
assume that
A458: front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C and
A459: front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ; ::_thesis: ( F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) )
consider i, j being Element of NAT  such that
A460: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len (F . k)) -' 1, Gauge (C,n) and
A461: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A156, A441, A450, A446, A457, A458, A459;
thus  F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) by A460, A461; ::_thesis: ( ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) )
A462: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by A461, FINSEQ_4:67;
A463: (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A445, A452, A461, FINSEQ_4:68;
A464: (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A443, A448, A461, FINSEQ_4:68;
hence  ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) by A442, A444, A455, A456, A451, A454, A460, A461, A463, A462, GOBRD13:def_7; ::_thesis: ( ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) )
thus  ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) by A442, A444, A455, A456, A449, A453, A460, A461, A464, A463, A462, GOBRD13:def_7; ::_thesis: ( ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) )
thus  ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) by A442, A444, A455, A456, A449, A453, A460, A461, A464, A463, A462, GOBRD13:def_7; ::_thesis: ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) )
thus  ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) by A442, A444, A455, A456, A451, A454, A460, A461, A464, A463, A462, GOBRD13:def_7; ::_thesis: verum
end;
end;
defpred S5[ Element of NAT ] means  for m being Element of NAT st m <= $1 holds
(F . $1) | m = F . m;
A465: S5[ 0 ]
proof
let m be Element of NAT ; ::_thesis: ( m <= 0 implies (F . 0) | m = F . m )
assume  m <= 0 ; ::_thesis: (F . 0) | m = F . m
then  0 = m ;
hence  (F . 0) | m = F . m by A155; ::_thesis: verum
end;
defpred S6[ Nat] means  ex w being Element of NAT st
( w = $1 & $1 >= 1 & ex m being Element of NAT st
( m in dom (F . w) & m <> len (F . w) & (F . w) /. m = (F . w) /. (len (F . w)) ) );
A466: S1[ 0 ,F . 0,F . (0 + 1)] by A156;
A467: for k, i1, i2, j1, j2 being Element of NAT st k > 1 & [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C holds
( F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) )
proof
let k, i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( k > 1 & [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) ) )
assume that
A468: k > 1 and
A469: [i1,j1] in Indices (Gauge (C,n)) and
A470: (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) and
A471: [i2,j2] in Indices (Gauge (C,n)) and
A472: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ; ::_thesis: ( not front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C or not front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C or ( F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) ) )
A473: len (F . k) = k by A176;
then A474: 1 <= (len (F . k)) -' 1 by A468, NAT_D:49;
 (len (F . k)) -' 1 <= len (F . k) by NAT_D:35;
then A475: (len (F . k)) -' 1 in dom (F . k) by A474, FINSEQ_3:25;
A476: i1 + 1 > i1 by NAT_1:13;
A477: F . k is_sequence_on Gauge (C,n) by A439;
A478: j1 + 1 > j1 by NAT_1:13;
A479: len (F . k) in dom (F . k) by A468, A473, FINSEQ_3:25;
A480: i2 + 1 > i2 by NAT_1:13;
A481: j2 + 1 > j2 by NAT_1:13;
A482: ((len (F . k)) -' 1) + 1 = len (F . k) by A468, A473, XREAL_1:235;
then A483: ((len (F . k)) -' 1) + (1 + 1) = (len (F . k)) + 1 ;
A484: left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C by A439, A474, A482;
hereby  ::_thesis: verum
assume that
A485: front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C and
A486: front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ( F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) )
consider i, j being Element of NAT  such that
A487: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len (F . k)) -' 1, Gauge (C,n) and
A488: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A156, A468, A477, A473, A484, A485, A486;
thus  F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) by A487, A488; ::_thesis: ( ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) )
A489: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by A488, FINSEQ_4:67;
A490: (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A472, A479, A488, FINSEQ_4:68;
A491: (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A470, A475, A488, FINSEQ_4:68;
hence  ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) by A469, A471, A482, A483, A478, A481, A487, A488, A490, A489, GOBRD13:def_8; ::_thesis: ( ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) )
thus  ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) by A469, A471, A482, A483, A476, A480, A487, A488, A491, A490, A489, GOBRD13:def_8; ::_thesis: ( ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) )
thus  ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) by A469, A471, A482, A483, A476, A480, A487, A488, A491, A490, A489, GOBRD13:def_8; ::_thesis: ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) )
thus  ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) by A469, A471, A482, A483, A478, A481, A487, A488, A491, A490, A489, GOBRD13:def_8; ::_thesis: verum
end;
end;
A492: for k, i1, i2, j1, j2 being Element of NAT st k > 1 & [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C holds
( F . (k + 1) turns_right (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) )
proof
let k, i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( k > 1 & [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) turns_right (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) ) )
assume that
A493: k > 1 and
A494: [i1,j1] in Indices (Gauge (C,n)) and
A495: (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) and
A496: [i2,j2] in Indices (Gauge (C,n)) and
A497: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ; ::_thesis: ( not front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C or ( F . (k + 1) turns_right (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) ) )
A498: len (F . k) = k by A176;
then A499: ((len (F . k)) -' 1) + 1 = len (F . k) by A493, XREAL_1:235;
then A500: ((len (F . k)) -' 1) + (1 + 1) = (len (F . k)) + 1 ;
A501: F . k is_sequence_on Gauge (C,n) by A439;
assume A502: front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ( F . (k + 1) turns_right (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) )
A503: 1 <= (len (F . k)) -' 1 by A493, A498, NAT_D:49;
then  left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C by A439, A499;
then consider i, j being Element of NAT  such that
A504: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len (F . k)) -' 1, Gauge (C,n) and
A505: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A156, A493, A501, A498, A502;
 len (F . k) in dom (F . k) by A493, A498, FINSEQ_3:25;
then A506: (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A497, A505, FINSEQ_4:68;
 (len (F . k)) -' 1 <= len (F . k) by NAT_D:35;
then  (len (F . k)) -' 1 in dom (F . k) by A503, FINSEQ_3:25;
then A507: (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A495, A505, FINSEQ_4:68;
thus  F . (k + 1) turns_right (len (F . k)) -' 1, Gauge (C,n) by A504, A505; ::_thesis: ( ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) )
A508: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by A505, FINSEQ_4:67;
A509: j2 + 1 > j2 by NAT_1:13;
A510: i2 + 1 > i2 by NAT_1:13;
A511: j1 + 1 > j1 by NAT_1:13;
hence  ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) by A494, A496, A499, A500, A509, A504, A505, A507, A506, A508, GOBRD13:def_6; ::_thesis: ( ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) )
A512: i1 + 1 > i1 by NAT_1:13;
hence  ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) by A494, A496, A499, A500, A510, A504, A505, A507, A506, A508, GOBRD13:def_6; ::_thesis: ( ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) )
thus  ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) by A494, A496, A499, A500, A512, A510, A504, A505, A507, A506, A508, GOBRD13:def_6; ::_thesis: ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) )
thus  ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) by A494, A496, A499, A500, A511, A509, A504, A505, A507, A506, A508, GOBRD13:def_6; ::_thesis: verum
end;
A513: for k being Element of NAT st k > 1 holds
( ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C implies F . (k + 1) turns_left k -' 1, Gauge (C,n) ) & ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) goes_straight k -' 1, Gauge (C,n) ) & ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) turns_right k -' 1, Gauge (C,n) ) )
proof
let k be Element of NAT ; ::_thesis: ( k > 1 implies ( ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C implies F . (k + 1) turns_left k -' 1, Gauge (C,n) ) & ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) goes_straight k -' 1, Gauge (C,n) ) & ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) turns_right k -' 1, Gauge (C,n) ) ) )
assume A514: k > 1 ; ::_thesis: ( ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C implies F . (k + 1) turns_left k -' 1, Gauge (C,n) ) & ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) goes_straight k -' 1, Gauge (C,n) ) & ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) turns_right k -' 1, Gauge (C,n) ) )
A515: F . k is_sequence_on Gauge (C,n) by A439;
A516: len (F . k) = k by A176;
then A517: ((len (F . k)) -' 1) + 1 = len (F . k) by A514, XREAL_1:235;
 1 <= (len (F . k)) -' 1 by A514, A516, NAT_D:49;
then  ex i1, j1, i2, j2 being Element of NAT st
( [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) & ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) ) by A515, A517, JORDAN8:3;
hence  ( ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C implies F . (k + 1) turns_left k -' 1, Gauge (C,n) ) & ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) goes_straight k -' 1, Gauge (C,n) ) & ( front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) turns_right k -' 1, Gauge (C,n) ) ) by A440, A467, A492, A514, A516; ::_thesis: verum
end;
A518: S1[1,F . 1,F . (1 + 1)] by A156;
A519: for k being Element of NAT ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
proof
let k be Element of NAT ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
A520: F . k is_sequence_on Gauge (C,n) by A439;
A521: len (F . k) = k by A176;
percases ( k < 1 or k = 1 or k > 1 ) by XXREAL_0:1;
supposeA522: k < 1 ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
take  X-SpanStart (C,n) ; ::_thesis: ex j being Element of NAT st
( [(X-SpanStart (C,n)),j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((X-SpanStart (C,n)),j))*> )
take  Y-SpanStart (C,n) ; ::_thesis: ( [(X-SpanStart (C,n)),(Y-SpanStart (C,n))] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))))*> )
thus  [(X-SpanStart (C,n)),(Y-SpanStart (C,n))] in Indices (Gauge (C,n)) by A1, JORDAN11:8; ::_thesis: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))))*>
 k = 0 by A522, NAT_1:14;
hence  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))))*> by A155, A466, FINSEQ_1:34; ::_thesis: verum
end;
supposeA523: k = 1 ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
take  (X-SpanStart (C,n)) -' 1 ; ::_thesis: ex j being Element of NAT st
( [((X-SpanStart (C,n)) -' 1),j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),j))*> )
take  Y-SpanStart (C,n) ; ::_thesis: ( [((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))))*> )
thus  [((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))] in Indices (Gauge (C,n)) by A1, JORDAN11:9; ::_thesis: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))))*>
thus  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))))*> by A466, A518, A523, FINSEQ_1:def_9; ::_thesis: verum
end;
supposeA524: k > 1 ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A525: ((len (F . k)) -' 1) + 1 = len (F . k) by A521, XREAL_1:235;
 1 <= (len (F . k)) -' 1 by A521, A524, NAT_D:49;
then consider i1, j1, i2, j2 being Element of NAT  such that
A526: [i1,j1] in Indices (Gauge (C,n)) and
A527: (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) and
A528: [i2,j2] in Indices (Gauge (C,n)) and
A529: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) and
A530: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A520, A525, JORDAN8:3;
now__::_thesis:_ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_F_._(k_+_1)_=_(F_._k)_^_<*((Gauge_(C,n))_*_(i,j))*>_)
percases ( ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) or ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ;
supposeA531: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_F_._(k_+_1)_=_(F_._k)_^_<*((Gauge_(C,n))_*_(i,j))*>_)
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A530;
supposeA532: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A533: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> by A440, A524, A526, A527, A528, A529, A531;
 [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A440, A524, A526, A527, A528, A529, A531, A532;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A533; ::_thesis: verum
end;
supposeA534: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A535: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> by A440, A524, A526, A527, A528, A529, A531;
 [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A440, A524, A526, A527, A528, A529, A531, A534;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A535; ::_thesis: verum
end;
supposeA536: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A537: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> by A440, A524, A526, A527, A528, A529, A531;
 [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A440, A524, A526, A527, A528, A529, A531, A536;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A537; ::_thesis: verum
end;
supposeA538: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A539: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> by A440, A524, A526, A527, A528, A529, A531;
 [(i2 + 1),j2] in Indices (Gauge (C,n)) by A440, A524, A526, A527, A528, A529, A531, A538;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A539; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
supposeA540: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_F_._(k_+_1)_=_(F_._k)_^_<*((Gauge_(C,n))_*_(i,j))*>_)
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A530;
supposeA541: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A542: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> by A467, A524, A526, A527, A528, A529, A540;
 [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A467, A524, A526, A527, A528, A529, A540, A541;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A542; ::_thesis: verum
end;
supposeA543: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A544: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> by A467, A524, A526, A527, A528, A529, A540;
 [(i2 + 1),j2] in Indices (Gauge (C,n)) by A467, A524, A526, A527, A528, A529, A540, A543;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A544; ::_thesis: verum
end;
supposeA545: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A546: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> by A467, A524, A526, A527, A528, A529, A540;
 [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A467, A524, A526, A527, A528, A529, A540, A545;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A546; ::_thesis: verum
end;
supposeA547: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A548: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> by A467, A524, A526, A527, A528, A529, A540;
 [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A467, A524, A526, A527, A528, A529, A540, A547;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A548; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
supposeA549: front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_F_._(k_+_1)_=_(F_._k)_^_<*((Gauge_(C,n))_*_(i,j))*>_)
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A530;
supposeA550: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A551: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> by A492, A524, A526, A527, A528, A529, A549;
 [(i2 + 1),j2] in Indices (Gauge (C,n)) by A492, A524, A526, A527, A528, A529, A549, A550;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A551; ::_thesis: verum
end;
supposeA552: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A553: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> by A492, A524, A526, A527, A528, A529, A549;
 [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A492, A524, A526, A527, A528, A529, A549, A552;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A553; ::_thesis: verum
end;
supposeA554: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A555: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> by A492, A524, A526, A527, A528, A529, A549;
 [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A492, A524, A526, A527, A528, A529, A549, A554;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A555; ::_thesis: verum
end;
supposeA556: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then A557: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> by A492, A524, A526, A527, A528, A529, A549;
 [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A492, A524, A526, A527, A528, A529, A549, A556;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A557; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
end;
end;
A558: for k being Element of NAT st S5[k] holds
S5[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S5[k] implies S5[k + 1] )
assume A559: for m being Element of NAT st m <= k holds
(F . k) | m = F . m ; ::_thesis: S5[k + 1]
let m be Element of NAT ; ::_thesis: ( m <= k + 1 implies (F . (k + 1)) | m = F . m )
assume A560: m <= k + 1 ; ::_thesis: (F . (k + 1)) | m = F . m
percases ( m < k + 1 or m = k + 1 ) by A560, XXREAL_0:1;
suppose m < k + 1 ; ::_thesis: (F . (k + 1)) | m = F . m
then A561: m <= k by NAT_1:13;
A562: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A519;
 len (F . k) = k by A176;
then  (F . (k + 1)) | m = (F . k) | m by A561, A562, FINSEQ_5:22;
hence  (F . (k + 1)) | m = F . m by A559, A561; ::_thesis: verum
end;
supposeA563: m = k + 1 ; ::_thesis: (F . (k + 1)) | m = F . m
 len (F . (k + 1)) = k + 1 by A176;
hence  (F . (k + 1)) | m = F . m by A563, FINSEQ_1:58; ::_thesis: verum
end;
end;
end;
A564: for k being Element of NAT holds S5[k] from NAT_1:sch_1(A465, A558);
A565: for j, k being Element of NAT st 1 <= j & j <= k holds
(F . k) /. j = (F . j) /. j
proof
let j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k implies (F . k) /. j = (F . j) /. j )
assume that
A566: 1 <= j and
A567: j <= k ; ::_thesis: (F . k) /. j = (F . j) /. j
 j <= len (F . k) by A176, A567;
then  len ((F . k) | j) = j by FINSEQ_1:59;
then A568: j in dom ((F . k) | j) by A566, FINSEQ_3:25;
 (F . k) | j = F . j by A564, A567;
hence  (F . k) /. j = (F . j) /. j by A568, FINSEQ_4:70; ::_thesis: verum
end;
defpred S7[ Element of NAT ] means F . $1 is unfolded ;
A569: for k being Element of NAT st S7[k] holds
S7[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S7[k] implies S7[k + 1] )
assume A570: F . k is unfolded ; ::_thesis: S7[k + 1]
A571: F . k is_sequence_on Gauge (C,n) by A439;
percases ( k <= 1 or k > 1 ) ;
suppose k <= 1 ; ::_thesis: S7[k + 1]
then  k + 1 <= 1 + 1 by XREAL_1:6;
then  len (F . (k + 1)) <= 2 by A176;
hence  S7[k + 1] by SPPOL_2:26; ::_thesis: verum
end;
supposeA572: k > 1 ; ::_thesis: S7[k + 1]
set m = k -' 1;
A573: (k -' 1) + 1 = k by A572, XREAL_1:235;
A574: len (F . k) = k by A176;
A575: 1 <= k -' 1 by A572, NAT_D:49;
then consider i1, j1, i2, j2 being Element of NAT  such that
A576: [i1,j1] in Indices (Gauge (C,n)) and
A577: (F . k) /. (k -' 1) = (Gauge (C,n)) * (i1,j1) and
A578: [i2,j2] in Indices (Gauge (C,n)) and
A579: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) and
A580: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A571, A573, A574, JORDAN8:3;
A581: LSeg ((F . k),(k -' 1)) = LSeg (((Gauge (C,n)) * (i1,j1)),((Gauge (C,n)) * (i2,j2))) by A575, A573, A574, A577, A579, TOPREAL1:def_3;
A582: 1 <= j2 by A578, MATRIX_1:38;
then A583: (j2 -' 1) + 1 = j2 by XREAL_1:235;
A584: 1 <= j1 by A576, MATRIX_1:38;
A585: 1 <= i2 by A578, MATRIX_1:38;
then A586: (i2 -' 1) + 1 = i2 by XREAL_1:235;
A587: i1 <= len (Gauge (C,n)) by A576, MATRIX_1:38;
A588: j2 <= width (Gauge (C,n)) by A578, MATRIX_1:38;
A589: 1 <= i1 by A576, MATRIX_1:38;
A590: j1 <= width (Gauge (C,n)) by A576, MATRIX_1:38;
A591: i2 <= len (Gauge (C,n)) by A578, MATRIX_1:38;
now__::_thesis:_S7[k_+_1]
percases ( ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) or ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ;
supposeA592: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: S7[k + 1]
now__::_thesis:_S7[k_+_1]
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A580;
supposeA593: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: S7[k + 1]
then  [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A440, A572, A574, A576, A577, A578, A579, A592;
then  1 <= i2 -' 1 by MATRIX_1:38;
then A594: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 -' 1),j2)))) = {((F . k) /. (len (F . k)))} by A579, A587, A584, A588, A586, A581, A593, GOBOARD7:16;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> by A440, A572, A574, A576, A577, A578, A579, A592, A593;
hence  S7[k + 1] by A570, A573, A574, A594, SPPOL_2:30; ::_thesis: verum
end;
supposeA595: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A440, A572, A574, A576, A577, A578, A579, A592;
then  j2 + 1 <= width (Gauge (C,n)) by MATRIX_1:38;
then A596: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 + 1))))) = {((F . k) /. (len (F . k)))} by A579, A589, A584, A591, A581, A595, GOBOARD7:18;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> by A440, A572, A574, A576, A577, A578, A579, A592, A595;
hence  S7[k + 1] by A570, A573, A574, A596, SPPOL_2:30; ::_thesis: verum
end;
supposeA597: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A440, A572, A574, A576, A577, A578, A579, A592;
then  1 <= j2 -' 1 by MATRIX_1:38;
then A598: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 -' 1))))) = {((F . k) /. (len (F . k)))} by A579, A587, A590, A585, A583, A581, A597, GOBOARD7:15;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> by A440, A572, A574, A576, A577, A578, A579, A592, A597;
hence  S7[k + 1] by A570, A573, A574, A598, SPPOL_2:30; ::_thesis: verum
end;
supposeA599: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: S7[k + 1]
then  [(i2 + 1),j2] in Indices (Gauge (C,n)) by A440, A572, A574, A576, A577, A578, A579, A592;
then  i2 + 1 <= len (Gauge (C,n)) by MATRIX_1:38;
then A600: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 + 1),j2)))) = {((F . k) /. (len (F . k)))} by A579, A589, A590, A582, A581, A599, GOBOARD7:17;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> by A440, A572, A574, A576, A577, A578, A579, A592, A599;
hence  S7[k + 1] by A570, A573, A574, A600, SPPOL_2:30; ::_thesis: verum
end;
end;
end;
hence  S7[k + 1] ; ::_thesis: verum
end;
supposeA601: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: S7[k + 1]
now__::_thesis:_S7[k_+_1]
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A580;
supposeA602: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A467, A572, A574, A576, A577, A578, A579, A601;
then A603: j2 + 1 <= width (Gauge (C,n)) by MATRIX_1:38;
 j2 + 1 = j1 + (1 + 1) by A602;
then A604: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 + 1))))) = {((F . k) /. (len (F . k)))} by A579, A589, A587, A584, A581, A602, A603, GOBOARD7:13;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> by A467, A572, A574, A576, A577, A578, A579, A601, A602;
hence  S7[k + 1] by A570, A573, A574, A604, SPPOL_2:30; ::_thesis: verum
end;
supposeA605: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [(i2 + 1),j2] in Indices (Gauge (C,n)) by A467, A572, A574, A576, A577, A578, A579, A601;
then A606: i2 + 1 <= len (Gauge (C,n)) by MATRIX_1:38;
 i2 + 1 = i1 + (1 + 1) by A605;
then A607: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 + 1),j2)))) = {((F . k) /. (len (F . k)))} by A579, A589, A584, A590, A581, A605, A606, GOBOARD7:14;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> by A467, A572, A574, A576, A577, A578, A579, A601, A605;
hence  S7[k + 1] by A570, A573, A574, A607, SPPOL_2:30; ::_thesis: verum
end;
supposeA608: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A467, A572, A574, A576, A577, A578, A579, A601;
then A609: 1 <= i2 -' 1 by MATRIX_1:38;
 ((i2 -' 1) + 1) + 1 = (i2 -' 1) + (1 + 1) ;
then A610: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 -' 1),j2)))) = {((F . k) /. (len (F . k)))} by A579, A587, A584, A590, A586, A581, A608, A609, GOBOARD7:14;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> by A467, A572, A574, A576, A577, A578, A579, A601, A608;
hence  S7[k + 1] by A570, A573, A574, A610, SPPOL_2:30; ::_thesis: verum
end;
supposeA611: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A467, A572, A574, A576, A577, A578, A579, A601;
then A612: 1 <= j2 -' 1 by MATRIX_1:38;
 ((j2 -' 1) + 1) + 1 = (j2 -' 1) + (1 + 1) ;
then A613: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 -' 1))))) = {((F . k) /. (len (F . k)))} by A579, A589, A587, A590, A583, A581, A611, A612, GOBOARD7:13;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> by A467, A572, A574, A576, A577, A578, A579, A601, A611;
hence  S7[k + 1] by A570, A573, A574, A613, SPPOL_2:30; ::_thesis: verum
end;
end;
end;
hence  S7[k + 1] ; ::_thesis: verum
end;
supposeA614: front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: S7[k + 1]
now__::_thesis:_S7[k_+_1]
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A580;
supposeA615: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: S7[k + 1]
then  [(i2 + 1),j2] in Indices (Gauge (C,n)) by A492, A572, A574, A576, A577, A578, A579, A614;
then  i2 + 1 <= len (Gauge (C,n)) by MATRIX_1:38;
then A616: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 + 1),j2)))) = {((F . k) /. (len (F . k)))} by A579, A589, A584, A588, A581, A615, GOBOARD7:15;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> by A492, A572, A574, A576, A577, A578, A579, A614, A615;
hence  S7[k + 1] by A570, A573, A574, A616, SPPOL_2:30; ::_thesis: verum
end;
supposeA617: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A492, A572, A574, A576, A577, A578, A579, A614;
then  1 <= j2 -' 1 by MATRIX_1:38;
then A618: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 -' 1))))) = {((F . k) /. (len (F . k)))} by A579, A589, A590, A591, A583, A581, A617, GOBOARD7:16;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> by A492, A572, A574, A576, A577, A578, A579, A614, A617;
hence  S7[k + 1] by A570, A573, A574, A618, SPPOL_2:30; ::_thesis: verum
end;
supposeA619: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A492, A572, A574, A576, A577, A578, A579, A614;
then  j2 + 1 <= width (Gauge (C,n)) by MATRIX_1:38;
then A620: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 + 1))))) = {((F . k) /. (len (F . k)))} by A579, A587, A584, A585, A581, A619, GOBOARD7:17;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> by A492, A572, A574, A576, A577, A578, A579, A614, A619;
hence  S7[k + 1] by A570, A573, A574, A620, SPPOL_2:30; ::_thesis: verum
end;
supposeA621: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: S7[k + 1]
then  [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A492, A572, A574, A576, A577, A578, A579, A614;
then  1 <= i2 -' 1 by MATRIX_1:38;
then A622: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 -' 1),j2)))) = {((F . k) /. (len (F . k)))} by A579, A587, A590, A582, A586, A581, A621, GOBOARD7:18;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> by A492, A572, A574, A576, A577, A578, A579, A614, A621;
hence  S7[k + 1] by A570, A573, A574, A622, SPPOL_2:30; ::_thesis: verum
end;
end;
end;
hence  S7[k + 1] ; ::_thesis: verum
end;
end;
end;
hence  S7[k + 1] ; ::_thesis: verum
end;
end;
end;
now__::_thesis:_ex_k_being_Element_of_NAT_st_
(_k_>=_1_&_ex_m_being_Element_of_NAT_st_
(_m_in_dom_(F_._k)_&_m_<>_len_(F_._k)_&_not_(F_._k)_/._m_<>_(F_._k)_/._(len_(F_._k))_)_)
defpred S8[ Element of NAT ] means F . $1 is one-to-one ;
assume A623: for k being Element of NAT st k >= 1 holds
for m being Element of NAT st m in dom (F . k) & m <> len (F . k) holds
(F . k) /. m <> (F . k) /. (len (F . k)) ; ::_thesis: contradiction
A624: for k being Element of NAT st S8[k] holds
S8[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S8[k] implies S8[k + 1] )
assume A625: F . k is one-to-one ; ::_thesis: S8[k + 1]
now__::_thesis:_for_n,_m_being_Element_of_NAT_st_n_in_dom_(F_._(k_+_1))_&_m_in_dom_(F_._(k_+_1))_&_(F_._(k_+_1))_/._n_=_(F_._(k_+_1))_/._m_holds_
n_=_m
let n, m be Element of NAT ; ::_thesis: ( n in dom (F . (k + 1)) & m in dom (F . (k + 1)) & (F . (k + 1)) /. n = (F . (k + 1)) /. m implies b1 = b2 )
assume that
A626: n in dom (F . (k + 1)) and
A627: m in dom (F . (k + 1)) and
A628: (F . (k + 1)) /. n = (F . (k + 1)) /. m ; ::_thesis: b1 = b2
A629: 1 <= n by A626, FINSEQ_3:25;
A630: m <= len (F . (k + 1)) by A627, FINSEQ_3:25;
A631: 1 <= m by A627, FINSEQ_3:25;
A632: n <= len (F . (k + 1)) by A626, FINSEQ_3:25;
A633: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A519;
A634: len (F . k) = k by A176;
A635: len (F . (k + 1)) = k + 1 by A176;
percases ( ( n <= k & m <= k ) or ( n = k + 1 & m <= k ) or ( n <= k & m = k + 1 ) or ( n = k + 1 & m = k + 1 ) ) by A632, A630, A635, NAT_1:8;
supposeA636: ( n <= k & m <= k ) ; ::_thesis: b1 = b2
then A637: m in dom (F . k) by A631, A634, FINSEQ_3:25;
then A638: (F . (k + 1)) /. m = (F . k) /. m by A633, FINSEQ_4:68;
A639: n in dom (F . k) by A629, A634, A636, FINSEQ_3:25;
then  (F . (k + 1)) /. n = (F . k) /. n by A633, FINSEQ_4:68;
hence  n = m by A625, A628, A639, A637, A638, PARTFUN2:10; ::_thesis: verum
end;
suppose ( n = k + 1 & m <= k ) ; ::_thesis: b1 = b2
hence  n = m by A623, A627, A628, A635, NAT_1:12; ::_thesis: verum
end;
suppose ( n <= k & m = k + 1 ) ; ::_thesis: b1 = b2
hence  n = m by A623, A626, A628, A635, NAT_1:12; ::_thesis: verum
end;
suppose ( n = k + 1 & m = k + 1 ) ; ::_thesis: b1 = b2
hence  n = m ; ::_thesis: verum
end;
end;
end;
hence  S8[k + 1] by PARTFUN2:9; ::_thesis: verum
end;
A640: S8[ 0 ] by A155;
A641: for k being Element of NAT holds S8[k] from NAT_1:sch_1(A640, A624);
A642: for k being Element of NAT holds card (rng (F . k)) = k
proof
let k be Element of NAT ; ::_thesis: card (rng (F . k)) = k
 F . k is one-to-one by A641;
hence  card (rng (F . k)) = len (F . k) by FINSEQ_4:62
.= k by A176 ;
::_thesis: verum
end;
reconsider k = ((len (Gauge (C,n))) * (width (Gauge (C,n)))) + 1 as Element of NAT ;
A643: k > (len (Gauge (C,n))) * (width (Gauge (C,n))) by NAT_1:13;
 F . k is_sequence_on Gauge (C,n) by A439;
then A644: card (rng (F . k)) <= card (Values (Gauge (C,n))) by GOBRD13:8, NAT_1:43;
 card (Values (Gauge (C,n))) <= (len (Gauge (C,n))) * (width (Gauge (C,n))) by MATRIX_1:46;
then  card (rng (F . k)) <= (len (Gauge (C,n))) * (width (Gauge (C,n))) by A644, XXREAL_0:2;
hence  contradiction by A642, A643; ::_thesis: verum
end;
then A645: ex k being Nat st S6[k] ;
consider k being Nat such that
A646: S6[k] and
A647: for l being Nat st S6[l] holds
k <= l from NAT_1:sch_5(A645);
reconsider k = k as Element of NAT by ORDINAL1:def_12;
consider m being Element of NAT  such that
A648: m in dom (F . k) and
A649: m <> len (F . k) and
A650: (F . k) /. m = (F . k) /. (len (F . k)) by A646;
A651: 1 <= m by A648, FINSEQ_3:25;
reconsider f = F . k as non empty FinSequence of (TOP-REAL 2) by A646;
A652: f is_sequence_on Gauge (C,n) by A439;
A653: m <= len f by A648, FINSEQ_3:25;
then A654: m < len f by A649, XXREAL_0:1;
then  1 < len f by A651, XXREAL_0:2;
then A655: len f >= 1 + 1 by NAT_1:13;
then A656: k >= 2 by A176;
A657: S7[ 0 ] by A155, CARD_1:27, SPPOL_2:26;
 for k being Element of NAT holds S7[k] from NAT_1:sch_1(A657, A569);
then reconsider f = f as non empty non constant special unfolded FinSequence of (TOP-REAL 2) by A652, A655, JORDAN8:4, JORDAN8:5;
set g = f /^ (m -' 1);
A658: m + 1 <= len f by A654, NAT_1:13;
A659: for h being non constant standard special_circular_sequence st L~ h c= L~ f holds
for Comp being Subset of (TOP-REAL 2) st Comp is_a_component_of (L~ h) ` holds
for n being Element of NAT st 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h holds
C meets Comp
proof
let h be non constant standard special_circular_sequence; ::_thesis: ( L~ h c= L~ f implies for Comp being Subset of (TOP-REAL 2) st Comp is_a_component_of (L~ h) ` holds
for n being Element of NAT st 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h holds
C meets Comp )
assume A660: L~ h c= L~ f ; ::_thesis: for Comp being Subset of (TOP-REAL 2) st Comp is_a_component_of (L~ h) ` holds
for n being Element of NAT st 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h holds
C meets Comp
let Comp be Subset of (TOP-REAL 2); ::_thesis: ( Comp is_a_component_of (L~ h) ` implies for n being Element of NAT st 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h holds
C meets Comp )
assume A661: Comp is_a_component_of (L~ h) ` ; ::_thesis: for n being Element of NAT st 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h holds
C meets Comp
let n be Element of NAT ; ::_thesis: ( 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h implies C meets Comp )
assume that
A662: 1 <= n and
A663: n + 1 <= len f and
A664: f /. n in Comp and
A665: not f /. n in L~ h ; ::_thesis: C meets Comp
set rc = (left_cell (f,n,(Gauge (C,n)))) \ (L~ h);
reconsider rc = (left_cell (f,n,(Gauge (C,n)))) \ (L~ h) as Subset of (TOP-REAL 2) ;
A666: Int (left_cell (f,n,(Gauge (C,n)))) c= left_cell (f,n,(Gauge (C,n))) by TOPS_1:16;
 f /. n in left_cell (f,n,(Gauge (C,n))) by A652, A662, A663, JORDAN9:8;
then  f /. n in rc by A665, XBOOLE_0:def_5;
then A667: rc meets Comp by A664, XBOOLE_0:3;
A668: rc = (left_cell (f,n,(Gauge (C,n)))) /\ ((L~ h) `) by SUBSET_1:13;
then A669: rc c= (L~ h) ` by XBOOLE_1:17;
 Int (left_cell (f,n,(Gauge (C,n)))) misses L~ f by A652, A662, A663, JORDAN9:15;
then  Int (left_cell (f,n,(Gauge (C,n)))) misses L~ h by A660, XBOOLE_1:63;
then A670: Int (left_cell (f,n,(Gauge (C,n)))) c= (L~ h) ` by SUBSET_1:23;
 rc c= left_cell (f,n,(Gauge (C,n))) by XBOOLE_1:36;
then A671: rc c= Cl (Int (left_cell (f,n,(Gauge (C,n))))) by A652, A662, A663, JORDAN9:11;
A672: rc meets C
proof
 left_cell (f,n,(Gauge (C,n))) meets C by A439, A662, A663;
then consider p being set  such that
A673: p in left_cell (f,n,(Gauge (C,n))) and
A674: p in C by XBOOLE_0:3;
reconsider p = p as Element of (TOP-REAL 2) by A673;
now__::_thesis:_ex_p_being_Element_of_(TOP-REAL_2)_st_
(_p_in_rc_&_p_in_C_)
take p = p; ::_thesis: ( p in rc & p in C )
now__::_thesis:_not_p_in_L~_h
assume  p in L~ h ; ::_thesis: contradiction
then consider j being Element of NAT  such that
A675: 1 <= j and
A676: j + 1 <= len f and
A677: p in LSeg (f,j) by A660, SPPOL_2:13;
 p in (right_cell (f,j,(Gauge (C,n)))) /\ (left_cell (f,j,(Gauge (C,n)))) by A439, A675, A676, A677, GOBRD13:29;
then A678: p in right_cell (f,j,(Gauge (C,n))) by XBOOLE_0:def_4;
 right_cell (f,j,(Gauge (C,n))) misses C by A439, A675, A676;
hence  contradiction by A674, A678, XBOOLE_0:3; ::_thesis: verum
end;
hence  p in rc by A673, XBOOLE_0:def_5; ::_thesis: p in C
thus  p in C by A674; ::_thesis: verum
end;
hence  rc meets C by XBOOLE_0:3; ::_thesis: verum
end;
 Int (left_cell (f,n,(Gauge (C,n)))) is convex by A652, A662, A663, JORDAN9:10;
then  rc is connected by A668, A670, A666, A671, CONNSP_1:18, XBOOLE_1:19;
then  rc c= Comp by A661, A667, A669, GOBOARD9:4;
hence  C meets Comp by A672, XBOOLE_1:63; ::_thesis: verum
end;
A679: for i being Element of NAT st 1 <= i & i + 1 <= len f holds
left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n)))))
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 1 <= len f implies left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n))))) )
assume that
A680: 1 <= i and
A681: i + 1 <= len f ; ::_thesis: left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n)))))
consider i1, j1, i2, j2 being Element of NAT  such that
A682: [i1,j1] in Indices (Gauge (C,n)) and
A683: f /. i = (Gauge (C,n)) * (i1,j1) and
A684: [i2,j2] in Indices (Gauge (C,n)) and
A685: f /. (i + 1) = (Gauge (C,n)) * (i2,j2) and
A686: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A652, A680, A681, JORDAN8:3;
A687: i1 <= len (Gauge (C,n)) by A682, MATRIX_1:38;
A688: i2 <= len (Gauge (C,n)) by A684, MATRIX_1:38;
A689: i1 + 1 > i1 by NAT_1:13;
A690: j1 <= width (Gauge (C,n)) by A682, MATRIX_1:38;
A691: j1 + 1 > j1 by NAT_1:13;
A692: j2 <= width (Gauge (C,n)) by A684, MATRIX_1:38;
A693: i2 + 1 > i2 by NAT_1:13;
A694: j2 + 1 > j2 by NAT_1:13;
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A686;
supposeA695: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n)))))
A696: i1 -' 1 <= len (Gauge (C,n)) by A687, NAT_D:44;
 left_cell (f,i,(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j1) by A652, A680, A681, A682, A683, A684, A685, A691, A694, A695, GOBRD13:def_3;
hence  left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n))))) by A690, A696, GOBRD11:35; ::_thesis: verum
end;
suppose ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n)))))
then  left_cell (f,i,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j1) by A652, A680, A681, A682, A683, A684, A685, A689, A693, GOBRD13:def_3;
hence  left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n))))) by A687, A690, GOBRD11:35; ::_thesis: verum
end;
supposeA697: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n)))))
A698: j2 -' 1 <= width (Gauge (C,n)) by A692, NAT_D:44;
 left_cell (f,i,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A652, A680, A681, A682, A683, A684, A685, A689, A693, A697, GOBRD13:def_3;
hence  left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n))))) by A688, A698, GOBRD11:35; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n)))))
then  left_cell (f,i,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j2) by A652, A680, A681, A682, A683, A684, A685, A691, A694, GOBRD13:def_3;
hence  left_cell (f,i,(Gauge (C,n))) = Cl (Int (left_cell (f,i,(Gauge (C,n))))) by A687, A692, GOBRD11:35; ::_thesis: verum
end;
end;
end;
 m -' 1 <= m by NAT_D:44;
then  m -' 1 < m + 1 by NAT_1:13;
then A699: m -' 1 < len f by A658, XXREAL_0:2;
then A700: len (f /^ (m -' 1)) = (len f) - (m -' 1) by RFINSEQ:def_1;
then  (m -' 1) - (m -' 1) < len (f /^ (m -' 1)) by A699, XREAL_1:9;
then reconsider g = f /^ (m -' 1) as non empty FinSequence of (TOP-REAL 2) by CARD_1:27;
 len g in dom g by FINSEQ_5:6;
then A701: g /. (len g) = f /. ((m -' 1) + (len g)) by FINSEQ_5:27
.= f /. (len f) by A700 ;
 (m + 1) - (m -' 1) <= len g by A658, A700, XREAL_1:9;
then A702: (m + 1) - (m - 1) <= len g by A651, XREAL_1:233;
then A703: ((1 + m) - m) + 1 <= len g ;
A704: g is_sequence_on Gauge (C,n) by A439, JORDAN8:2;
then A705: g is standard by JORDAN8:4;
A706: not g is constant
proof
take  1 ; :: according to SEQM_3:def_10 ::_thesis: ex b1 being Element of NAT st
( 1 in dom g & b1 in dom g & not g . 1 = g . b1 )
take  2 ; ::_thesis: ( 1 in dom g & 2 in dom g & not g . 1 = g . 2 )
thus A707: 1 in dom g by FINSEQ_5:6; ::_thesis: ( 2 in dom g & not g . 1 = g . 2 )
thus A708: 2 in dom g by A702, FINSEQ_3:25; ::_thesis: not g . 1 = g . 2
then  g /. 1 <> g /. (1 + 1) by A705, FINSEQ_5:6, GOBOARD7:29;
then  g . 1 <> g /. (1 + 1) by A707, PARTFUN1:def_6;
hence  not g . 1 = g . 2 by A708, PARTFUN1:def_6; ::_thesis: verum
end;
A709: len (F . k) = k by A176;
A710: for j, i being Element of NAT st 1 <= i & i < len g & 1 <= j & j < len g & g /. i = g /. j holds
i = j
proof
let j, i be Element of NAT ; ::_thesis: ( 1 <= i & i < len g & 1 <= j & j < len g & g /. i = g /. j implies i = j )
assume that
A711: 1 <= i and
A712: i < len g and
A713: 1 <= j and
A714: j < len g and
A715: g /. i = g /. j and
A716: i <> j ; ::_thesis: contradiction
A717: i in dom g by A711, A712, FINSEQ_3:25;
then A718: g /. i = f /. ((m -' 1) + i) by FINSEQ_5:27;
A719: j in dom g by A713, A714, FINSEQ_3:25;
then A720: g /. j = f /. ((m -' 1) + j) by FINSEQ_5:27;
percases ( i < j or j < i ) by A716, XXREAL_0:1;
supposeA721: i < j ; ::_thesis: contradiction
set l = (m -' 1) + j;
set m9 = (m -' 1) + i;
A722: (m -' 1) + i < (m -' 1) + j by A721, XREAL_1:6;
A723: len (F . ((m -' 1) + j)) = (m -' 1) + j by A176;
A724: (m -' 1) + j < k by A709, A700, A714, XREAL_1:20;
then A725: f | ((m -' 1) + j) = F . ((m -' 1) + j) by A564;
 0 + j <= (m -' 1) + j by XREAL_1:6;
then A726: 1 <= (m -' 1) + j by A713, XXREAL_0:2;
then  (m -' 1) + j in dom (F . ((m -' 1) + j)) by A723, FINSEQ_3:25;
then A727: (F . ((m -' 1) + j)) /. ((m -' 1) + j) = f /. ((m -' 1) + j) by A725, FINSEQ_4:70;
 0 + i <= (m -' 1) + i by XREAL_1:6;
then  1 <= (m -' 1) + i by A711, XXREAL_0:2;
then A728: (m -' 1) + i in dom (F . ((m -' 1) + j)) by A722, A723, FINSEQ_3:25;
then  (F . ((m -' 1) + j)) /. ((m -' 1) + i) = f /. ((m -' 1) + i) by A725, FINSEQ_4:70;
hence  contradiction by A647, A715, A718, A719, A722, A724, A726, A723, A728, A727, FINSEQ_5:27; ::_thesis: verum
end;
supposeA729: j < i ; ::_thesis: contradiction
set l = (m -' 1) + i;
set m9 = (m -' 1) + j;
A730: (m -' 1) + j < (m -' 1) + i by A729, XREAL_1:6;
A731: len (F . ((m -' 1) + i)) = (m -' 1) + i by A176;
A732: (m -' 1) + i < k by A709, A700, A712, XREAL_1:20;
then A733: f | ((m -' 1) + i) = F . ((m -' 1) + i) by A564;
 0 + i <= (m -' 1) + i by XREAL_1:6;
then A734: 1 <= (m -' 1) + i by A711, XXREAL_0:2;
then  (m -' 1) + i in dom (F . ((m -' 1) + i)) by A731, FINSEQ_3:25;
then A735: (F . ((m -' 1) + i)) /. ((m -' 1) + i) = f /. ((m -' 1) + i) by A733, FINSEQ_4:70;
 0 + j <= (m -' 1) + j by XREAL_1:6;
then  1 <= (m -' 1) + j by A713, XXREAL_0:2;
then A736: (m -' 1) + j in dom (F . ((m -' 1) + i)) by A730, A731, FINSEQ_3:25;
then  (F . ((m -' 1) + i)) /. ((m -' 1) + j) = f /. ((m -' 1) + j) by A733, FINSEQ_4:70;
hence  contradiction by A647, A715, A717, A720, A730, A732, A734, A731, A736, A735, FINSEQ_5:27; ::_thesis: verum
end;
end;
end;
 1 in dom g by FINSEQ_5:6;
then A737: g /. 1 = f /. ((m -' 1) + 1) by FINSEQ_5:27
.= f /. m by A651, XREAL_1:235 ;
A738: for j, i being Element of NAT st 1 < i & i < j & j <= len g holds
g /. i <> g /. j
proof
let j, i be Element of NAT ; ::_thesis: ( 1 < i & i < j & j <= len g implies g /. i <> g /. j )
assume that
A739: 1 < i and
A740: i < j and
A741: j <= len g and
A742: g /. i = g /. j ; ::_thesis: contradiction
A743: 1 < j by A739, A740, XXREAL_0:2;
A744: i < len g by A740, A741, XXREAL_0:2;
then A745: 1 < len g by A739, XXREAL_0:2;
percases ( j <> len g or j = len g ) ;
suppose j <> len g ; ::_thesis: contradiction
then  j < len g by A741, XXREAL_0:1;
hence  contradiction by A710, A739, A740, A742, A743, A744; ::_thesis: verum
end;
suppose j = len g ; ::_thesis: contradiction
hence  contradiction by A650, A737, A701, A710, A739, A740, A742, A745; ::_thesis: verum
end;
end;
end;
A746: for j, i being Element of NAT st 1 <= i & i < j & j < len g holds
g /. i <> g /. j
proof
let j, i be Element of NAT ; ::_thesis: ( 1 <= i & i < j & j < len g implies g /. i <> g /. j )
assume that
A747: 1 <= i and
A748: i < j and
A749: j < len g and
A750: g /. i = g /. j ; ::_thesis: contradiction
A751: i < len g by A748, A749, XXREAL_0:2;
 1 < j by A747, A748, XXREAL_0:2;
hence  contradiction by A710, A747, A748, A749, A750, A751; ::_thesis: verum
end;
 g is s.c.c.
proof
let i be Element of NAT ; :: according to GOBOARD5:def_4 ::_thesis: for b1 being Element of NAT holds
( b1 <= i + 1 or ( ( i <= 1 or len g <= b1 ) & len g <= b1 + 1 ) or LSeg (g,i) misses LSeg (g,b1) )
let j be Element of NAT ; ::_thesis: ( j <= i + 1 or ( ( i <= 1 or len g <= j ) & len g <= j + 1 ) or LSeg (g,i) misses LSeg (g,j) )
assume that
A752: i + 1 < j and
A753: ( ( i > 1 & j < len g ) or j + 1 < len g ) ; ::_thesis: LSeg (g,i) misses LSeg (g,j)
A754: 1 < j by A752, NAT_1:12;
A755: 1 <= i + 1 by NAT_1:12;
A756: j <= j + 1 by NAT_1:12;
then A757: i + 1 < j + 1 by A752, XXREAL_0:2;
 i < j by A752, NAT_1:13;
then A758: i < j + 1 by A756, XXREAL_0:2;
percases ( ( i > 1 & j < len g ) or ( i = 0 & j + 1 < len g ) or ( 1 <= i & j + 1 < len g ) ) by A753, NAT_1:14;
supposeA759: ( i > 1 & j < len g ) ; ::_thesis: LSeg (g,i) misses LSeg (g,j)
then A760: i + 1 < len g by A752, XXREAL_0:2;
then A761: LSeg (g,i) = LSeg ((g /. i),(g /. (i + 1))) by A759, TOPREAL1:def_3;
A762: i < len g by A760, NAT_1:13;
consider i1, j1, i2, j2 being Element of NAT  such that
A763: [i1,j1] in Indices (Gauge (C,n)) and
A764: g /. i = (Gauge (C,n)) * (i1,j1) and
A765: [i2,j2] in Indices (Gauge (C,n)) and
A766: g /. (i + 1) = (Gauge (C,n)) * (i2,j2) and
A767: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A704, A759, A760, JORDAN8:3;
A768: 1 <= i1 by A763, MATRIX_1:38;
A769: j2 <= width (Gauge (C,n)) by A765, MATRIX_1:38;
A770: 1 <= i2 by A765, MATRIX_1:38;
A771: i1 <= len (Gauge (C,n)) by A763, MATRIX_1:38;
A772: 1 <= j2 by A765, MATRIX_1:38;
A773: j1 <= width (Gauge (C,n)) by A763, MATRIX_1:38;
A774: i2 <= len (Gauge (C,n)) by A765, MATRIX_1:38;
A775: 1 <= j1 by A763, MATRIX_1:38;
A776: 1 < i + 1 by A759, NAT_1:13;
A777: j + 1 <= len g by A759, NAT_1:13;
then A778: LSeg (g,j) = LSeg ((g /. j),(g /. (j + 1))) by A754, TOPREAL1:def_3;
consider i19, j19, i29, j29 being Element of NAT  such that
A779: [i19,j19] in Indices (Gauge (C,n)) and
A780: g /. j = (Gauge (C,n)) * (i19,j19) and
A781: [i29,j29] in Indices (Gauge (C,n)) and
A782: g /. (j + 1) = (Gauge (C,n)) * (i29,j29) and
A783: ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A704, A754, A777, JORDAN8:3;
A784: 1 <= i19 by A779, MATRIX_1:38;
A785: j29 <= width (Gauge (C,n)) by A781, MATRIX_1:38;
A786: j19 <= width (Gauge (C,n)) by A779, MATRIX_1:38;
A787: 1 <= j29 by A781, MATRIX_1:38;
A788: 1 <= j19 by A779, MATRIX_1:38;
A789: i29 <= len (Gauge (C,n)) by A781, MATRIX_1:38;
A790: i19 <= len (Gauge (C,n)) by A779, MATRIX_1:38;
assume  (LSeg (g,i)) /\ (LSeg (g,j)) <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction
then A791: LSeg (g,i) meets LSeg (g,j) by XBOOLE_0:def_7;
A792: 1 <= i29 by A781, MATRIX_1:38;
now__::_thesis:_contradiction
percases ( ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 = j29 + 1 ) ) by A767, A783;
supposeA793: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then A794: i1 = i19 by A761, A764, A766, A768, A771, A775, A769, A778, A780, A782, A784, A790, A788, A785, A791, GOBOARD7:19;
now__::_thesis:_contradiction
percases ( j1 = j19 or j1 = j19 + 1 or j1 + 1 = j19 ) by A761, A764, A766, A768, A771, A775, A769, A778, A780, A782, A784, A790, A788, A785, A791, A793, GOBOARD7:22;
suppose j1 = j19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A794; ::_thesis: verum
end;
suppose j1 = j19 + 1 ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A793, A794; ::_thesis: verum
end;
suppose j1 + 1 = j19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A793, A794; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA795: ( i1 = i2 & j1 + 1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i1 = i19 & j1 = j19 ) or ( i1 = i19 & j1 + 1 = j19 ) or ( i1 = i19 + 1 & j1 = j19 ) or ( i1 = i19 + 1 & j1 + 1 = j19 ) ) by A761, A764, A766, A768, A771, A775, A769, A778, A780, A782, A784, A788, A786, A789, A791, A795, GOBOARD7:21;
suppose ( i1 = i19 & j1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780; ::_thesis: verum
end;
suppose ( i1 = i19 & j1 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A752, A759, A776, A766, A780, A795; ::_thesis: verum
end;
suppose ( i1 = i19 + 1 & j1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A795; ::_thesis: verum
end;
suppose ( i1 = i19 + 1 & j1 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A795; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA796: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i1 = i29 & j19 = j1 ) or ( i1 = i29 & j1 + 1 = j19 ) or ( i1 = i29 + 1 & j19 = j1 ) or ( i1 = i29 + 1 & j1 + 1 = j19 ) ) by A761, A764, A766, A768, A771, A775, A769, A778, A780, A782, A790, A788, A786, A792, A791, A796, GOBOARD7:21;
suppose ( i1 = i29 & j19 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A796; ::_thesis: verum
end;
suppose ( i1 = i29 & j1 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A796; ::_thesis: verum
end;
suppose ( i1 = i29 + 1 & j19 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A796; ::_thesis: verum
end;
suppose ( i1 = i29 + 1 & j1 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A796; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA797: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then A798: i1 = i19 by A761, A764, A766, A768, A771, A775, A769, A778, A780, A782, A784, A790, A786, A787, A791, GOBOARD7:19;
now__::_thesis:_contradiction
percases ( j1 = j29 or j1 = j29 + 1 or j1 + 1 = j29 ) by A761, A764, A766, A768, A771, A775, A769, A778, A780, A782, A784, A790, A786, A787, A791, A797, GOBOARD7:22;
suppose j1 = j29 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A797, A798; ::_thesis: verum
end;
suppose j1 = j29 + 1 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A797, A798; ::_thesis: verum
end;
suppose j1 + 1 = j29 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A797, A798; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA799: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i19 = i1 & j1 = j19 ) or ( i19 = i1 & j19 + 1 = j1 ) or ( i19 = i1 + 1 & j1 = j19 ) or ( i19 = i1 + 1 & j19 + 1 = j1 ) ) by A761, A764, A766, A768, A775, A773, A774, A778, A780, A782, A784, A790, A788, A785, A791, A799, GOBOARD7:21;
suppose ( i19 = i1 & j1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780; ::_thesis: verum
end;
suppose ( i19 = i1 & j19 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A799; ::_thesis: verum
end;
suppose ( i19 = i1 + 1 & j1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A799; ::_thesis: verum
end;
suppose ( i19 = i1 + 1 & j19 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A799; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA800: ( i1 + 1 = i2 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then A801: j1 = j19 by A761, A764, A766, A768, A775, A773, A774, A778, A780, A782, A784, A788, A786, A789, A791, GOBOARD7:20;
now__::_thesis:_contradiction
percases ( i1 = i19 or i1 = i19 + 1 or i1 + 1 = i19 ) by A761, A764, A766, A768, A775, A773, A774, A778, A780, A782, A784, A788, A786, A789, A791, A800, GOBOARD7:23;
suppose i1 = i19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A801; ::_thesis: verum
end;
suppose i1 = i19 + 1 ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A800, A801; ::_thesis: verum
end;
suppose i1 + 1 = i19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A800, A801; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA802: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then A803: j1 = j19 by A761, A764, A766, A768, A775, A773, A774, A778, A780, A782, A790, A788, A786, A792, A791, GOBOARD7:20;
now__::_thesis:_contradiction
percases ( i1 = i29 or i1 = i29 + 1 or i1 + 1 = i29 ) by A761, A764, A766, A768, A775, A773, A774, A778, A780, A782, A790, A788, A786, A792, A791, A802, GOBOARD7:23;
suppose i1 = i29 ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A802, A803; ::_thesis: verum
end;
suppose i1 = i29 + 1 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A802, A803; ::_thesis: verum
end;
suppose i1 + 1 = i29 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A802, A803; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA804: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i19 = i1 & j1 = j29 ) or ( i19 = i1 & j29 + 1 = j1 ) or ( i19 = i1 + 1 & j1 = j29 ) or ( i19 = i1 + 1 & j29 + 1 = j1 ) ) by A761, A764, A766, A768, A775, A773, A774, A778, A780, A782, A784, A790, A786, A787, A791, A804, GOBOARD7:21;
suppose ( i19 = i1 & j1 = j29 ) ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A804; ::_thesis: verum
end;
suppose ( i19 = i1 & j29 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A804; ::_thesis: verum
end;
suppose ( i19 = i1 + 1 & j1 = j29 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A804; ::_thesis: verum
end;
suppose ( i19 = i1 + 1 & j29 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A804; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA805: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i19 = i2 & j19 = j1 ) or ( i19 = i2 & j19 + 1 = j1 ) or ( i19 = i2 + 1 & j19 = j1 ) or ( i19 = i2 + 1 & j19 + 1 = j1 ) ) by A761, A764, A766, A771, A775, A773, A770, A778, A780, A782, A784, A790, A788, A785, A791, A805, GOBOARD7:21;
suppose ( i19 = i2 & j19 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A805; ::_thesis: verum
end;
suppose ( i19 = i2 & j19 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A805; ::_thesis: verum
end;
suppose ( i19 = i2 + 1 & j19 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A805; ::_thesis: verum
end;
suppose ( i19 = i2 + 1 & j19 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A805; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA806: ( i1 = i2 + 1 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then A807: j1 = j19 by A761, A764, A766, A771, A775, A773, A770, A778, A780, A782, A784, A788, A786, A789, A791, GOBOARD7:20;
now__::_thesis:_contradiction
percases ( i2 = i19 or i2 = i19 + 1 or i2 + 1 = i19 ) by A761, A764, A766, A771, A775, A773, A770, A778, A780, A782, A784, A788, A786, A789, A791, A806, GOBOARD7:23;
suppose i2 = i19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A806, A807; ::_thesis: verum
end;
suppose i2 = i19 + 1 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A806, A807; ::_thesis: verum
end;
suppose i2 + 1 = i19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A806, A807; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA808: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then A809: j1 = j19 by A761, A764, A766, A771, A775, A773, A770, A778, A780, A782, A790, A788, A786, A792, A791, GOBOARD7:20;
now__::_thesis:_contradiction
percases ( i2 = i29 or i2 = i29 + 1 or i2 + 1 = i29 ) by A761, A764, A766, A771, A775, A773, A770, A778, A780, A782, A790, A788, A786, A792, A791, A808, GOBOARD7:23;
suppose i2 = i29 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A808, A809; ::_thesis: verum
end;
suppose i2 = i29 + 1 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A808, A809; ::_thesis: verum
end;
suppose i2 + 1 = i29 ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A808, A809; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA810: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i19 = i2 & j29 = j1 ) or ( i19 = i2 & j29 + 1 = j1 ) or ( i19 = i2 + 1 & j29 = j1 ) or ( i19 = i2 + 1 & j29 + 1 = j1 ) ) by A761, A764, A766, A771, A775, A773, A770, A778, A780, A782, A784, A790, A786, A787, A791, A810, GOBOARD7:21;
suppose ( i19 = i2 & j29 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A810; ::_thesis: verum
end;
suppose ( i19 = i2 & j29 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A810; ::_thesis: verum
end;
suppose ( i19 = i2 + 1 & j29 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A810; ::_thesis: verum
end;
suppose ( i19 = i2 + 1 & j29 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A810; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA811: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then A812: i1 = i19 by A761, A764, A766, A768, A771, A773, A772, A778, A780, A782, A784, A790, A788, A785, A791, GOBOARD7:19;
now__::_thesis:_contradiction
percases ( j2 = j19 or j2 = j19 + 1 or j2 + 1 = j19 ) by A761, A764, A766, A768, A771, A773, A772, A778, A780, A782, A784, A790, A788, A785, A791, A811, GOBOARD7:22;
suppose j2 = j19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A811, A812; ::_thesis: verum
end;
suppose j2 = j19 + 1 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A811, A812; ::_thesis: verum
end;
suppose j2 + 1 = j19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A811, A812; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA813: ( i1 = i2 & j1 = j2 + 1 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i1 = i19 & j2 = j19 ) or ( i1 = i19 & j2 + 1 = j19 ) or ( i1 = i19 + 1 & j2 = j19 ) or ( i1 = i19 + 1 & j2 + 1 = j19 ) ) by A761, A764, A766, A768, A771, A773, A772, A778, A780, A782, A784, A788, A786, A789, A791, A813, GOBOARD7:21;
suppose ( i1 = i19 & j2 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A813; ::_thesis: verum
end;
suppose ( i1 = i19 & j2 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A813; ::_thesis: verum
end;
suppose ( i1 = i19 + 1 & j2 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A813; ::_thesis: verum
end;
suppose ( i1 = i19 + 1 & j2 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A813; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA814: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i1 = i29 & j2 = j19 ) or ( i1 = i29 & j2 + 1 = j19 ) or ( i1 = i29 + 1 & j2 = j19 ) or ( i1 = i29 + 1 & j2 + 1 = j19 ) ) by A761, A764, A766, A768, A771, A773, A772, A778, A780, A782, A790, A788, A786, A792, A791, A814, GOBOARD7:21;
suppose ( i1 = i29 & j2 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A814; ::_thesis: verum
end;
suppose ( i1 = i29 & j2 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A814; ::_thesis: verum
end;
suppose ( i1 = i29 + 1 & j2 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A814; ::_thesis: verum
end;
suppose ( i1 = i29 + 1 & j2 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A759, A762, A764, A780, A814; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA815: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then A816: i1 = i19 by A761, A764, A766, A768, A771, A773, A772, A778, A780, A782, A784, A790, A786, A787, A791, GOBOARD7:19;
now__::_thesis:_contradiction
percases ( j2 = j29 or j2 = j29 + 1 or j2 + 1 = j29 ) by A761, A764, A766, A768, A771, A773, A772, A778, A780, A782, A784, A790, A786, A787, A791, A815, GOBOARD7:22;
suppose j2 = j29 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A776, A766, A777, A782, A815, A816; ::_thesis: verum
end;
suppose j2 = j29 + 1 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A759, A760, A766, A780, A815, A816; ::_thesis: verum
end;
suppose j2 + 1 = j29 ; ::_thesis: contradiction
hence  contradiction by A738, A758, A759, A764, A777, A782, A815, A816; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
suppose ( i = 0 & j + 1 < len g ) ; ::_thesis: LSeg (g,i) misses LSeg (g,j)
then  LSeg (g,i) = {} by TOPREAL1:def_3;
hence  (LSeg (g,i)) /\ (LSeg (g,j)) = {} ; :: according to XBOOLE_0:def_7 ::_thesis: verum
end;
supposeA817: ( 1 <= i & j + 1 < len g ) ; ::_thesis: LSeg (g,i) misses LSeg (g,j)
then consider i19, j19, i29, j29 being Element of NAT  such that
A818: [i19,j19] in Indices (Gauge (C,n)) and
A819: g /. j = (Gauge (C,n)) * (i19,j19) and
A820: [i29,j29] in Indices (Gauge (C,n)) and
A821: g /. (j + 1) = (Gauge (C,n)) * (i29,j29) and
A822: ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A704, A754, JORDAN8:3;
A823: 1 <= i19 by A818, MATRIX_1:38;
A824: j29 <= width (Gauge (C,n)) by A820, MATRIX_1:38;
A825: 1 <= i29 by A820, MATRIX_1:38;
A826: i19 <= len (Gauge (C,n)) by A818, MATRIX_1:38;
A827: 1 <= j29 by A820, MATRIX_1:38;
A828: j19 <= width (Gauge (C,n)) by A818, MATRIX_1:38;
A829: i29 <= len (Gauge (C,n)) by A820, MATRIX_1:38;
A830: 1 <= j19 by A818, MATRIX_1:38;
assume  (LSeg (g,i)) /\ (LSeg (g,j)) <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction
then A831: LSeg (g,i) meets LSeg (g,j) by XBOOLE_0:def_7;
A832: 1 < i + 1 by A817, NAT_1:13;
A833: j < len g by A817, NAT_1:12;
A834: i + 1 < len g by A757, A817, XXREAL_0:2;
then A835: LSeg (g,i) = LSeg ((g /. i),(g /. (i + 1))) by A817, TOPREAL1:def_3;
A836: i < len g by A834, NAT_1:13;
consider i1, j1, i2, j2 being Element of NAT  such that
A837: [i1,j1] in Indices (Gauge (C,n)) and
A838: g /. i = (Gauge (C,n)) * (i1,j1) and
A839: [i2,j2] in Indices (Gauge (C,n)) and
A840: g /. (i + 1) = (Gauge (C,n)) * (i2,j2) and
A841: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A704, A817, A834, JORDAN8:3;
A842: 1 <= i1 by A837, MATRIX_1:38;
A843: j2 <= width (Gauge (C,n)) by A839, MATRIX_1:38;
A844: j1 <= width (Gauge (C,n)) by A837, MATRIX_1:38;
A845: 1 <= j2 by A839, MATRIX_1:38;
A846: 1 <= j1 by A837, MATRIX_1:38;
A847: i2 <= len (Gauge (C,n)) by A839, MATRIX_1:38;
A848: i1 <= len (Gauge (C,n)) by A837, MATRIX_1:38;
A849: LSeg (g,j) = LSeg ((g /. j),(g /. (j + 1))) by A754, A817, TOPREAL1:def_3;
A850: 1 <= i2 by A839, MATRIX_1:38;
now__::_thesis:_contradiction
percases ( ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 = j29 + 1 ) ) by A841, A822;
supposeA851: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then A852: i1 = i19 by A835, A838, A840, A842, A848, A846, A843, A849, A819, A821, A823, A826, A830, A824, A831, GOBOARD7:19;
now__::_thesis:_contradiction
percases ( j1 = j19 or j1 = j19 + 1 or j1 + 1 = j19 ) by A835, A838, A840, A842, A848, A846, A843, A849, A819, A821, A823, A826, A830, A824, A831, A851, GOBOARD7:22;
suppose j1 = j19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A852; ::_thesis: verum
end;
suppose j1 = j19 + 1 ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A851, A852; ::_thesis: verum
end;
suppose j1 + 1 = j19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A851, A852; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA853: ( i1 = i2 & j1 + 1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i1 = i19 & j1 = j19 ) or ( i1 = i19 & j1 + 1 = j19 ) or ( i1 = i19 + 1 & j1 = j19 ) or ( i1 = i19 + 1 & j1 + 1 = j19 ) ) by A835, A838, A840, A842, A848, A846, A843, A849, A819, A821, A823, A830, A828, A829, A831, A853, GOBOARD7:21;
suppose ( i1 = i19 & j1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819; ::_thesis: verum
end;
suppose ( i1 = i19 & j1 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A752, A832, A833, A840, A819, A853; ::_thesis: verum
end;
suppose ( i1 = i19 + 1 & j1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A853; ::_thesis: verum
end;
suppose ( i1 = i19 + 1 & j1 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A853; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA854: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i1 = i29 & j19 = j1 ) or ( i1 = i29 & j1 + 1 = j19 ) or ( i1 = i29 + 1 & j19 = j1 ) or ( i1 = i29 + 1 & j1 + 1 = j19 ) ) by A835, A838, A840, A842, A848, A846, A843, A849, A819, A821, A826, A830, A828, A825, A831, A854, GOBOARD7:21;
suppose ( i1 = i29 & j19 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A854; ::_thesis: verum
end;
suppose ( i1 = i29 & j1 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A854; ::_thesis: verum
end;
suppose ( i1 = i29 + 1 & j19 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A854; ::_thesis: verum
end;
suppose ( i1 = i29 + 1 & j1 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A854; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA855: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then A856: i1 = i19 by A835, A838, A840, A842, A848, A846, A843, A849, A819, A821, A823, A826, A828, A827, A831, GOBOARD7:19;
now__::_thesis:_contradiction
percases ( j1 = j29 or j1 = j29 + 1 or j1 + 1 = j29 ) by A835, A838, A840, A842, A848, A846, A843, A849, A819, A821, A823, A826, A828, A827, A831, A855, GOBOARD7:22;
suppose j1 = j29 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A855, A856; ::_thesis: verum
end;
suppose j1 = j29 + 1 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A855, A856; ::_thesis: verum
end;
suppose j1 + 1 = j29 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A855, A856; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA857: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i19 = i1 & j1 = j19 ) or ( i19 = i1 & j19 + 1 = j1 ) or ( i19 = i1 + 1 & j1 = j19 ) or ( i19 = i1 + 1 & j19 + 1 = j1 ) ) by A835, A838, A840, A842, A846, A844, A847, A849, A819, A821, A823, A826, A830, A824, A831, A857, GOBOARD7:21;
suppose ( i19 = i1 & j1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819; ::_thesis: verum
end;
suppose ( i19 = i1 & j19 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A857; ::_thesis: verum
end;
suppose ( i19 = i1 + 1 & j1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A857; ::_thesis: verum
end;
suppose ( i19 = i1 + 1 & j19 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A857; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA858: ( i1 + 1 = i2 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then A859: j1 = j19 by A835, A838, A840, A842, A846, A844, A847, A849, A819, A821, A823, A830, A828, A829, A831, GOBOARD7:20;
now__::_thesis:_contradiction
percases ( i1 = i19 or i1 = i19 + 1 or i1 + 1 = i19 ) by A835, A838, A840, A842, A846, A844, A847, A849, A819, A821, A823, A830, A828, A829, A831, A858, GOBOARD7:23;
suppose i1 = i19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A859; ::_thesis: verum
end;
suppose i1 = i19 + 1 ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A858, A859; ::_thesis: verum
end;
suppose i1 + 1 = i19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A858, A859; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA860: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then A861: j1 = j19 by A835, A838, A840, A842, A846, A844, A847, A849, A819, A821, A826, A830, A828, A825, A831, GOBOARD7:20;
now__::_thesis:_contradiction
percases ( i1 = i29 or i1 = i29 + 1 or i1 + 1 = i29 ) by A835, A838, A840, A842, A846, A844, A847, A849, A819, A821, A826, A830, A828, A825, A831, A860, GOBOARD7:23;
suppose i1 = i29 ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A860, A861; ::_thesis: verum
end;
suppose i1 = i29 + 1 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A860, A861; ::_thesis: verum
end;
suppose i1 + 1 = i29 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A860, A861; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA862: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i19 = i1 & j1 = j29 ) or ( i19 = i1 & j29 + 1 = j1 ) or ( i19 = i1 + 1 & j1 = j29 ) or ( i19 = i1 + 1 & j29 + 1 = j1 ) ) by A835, A838, A840, A842, A846, A844, A847, A849, A819, A821, A823, A826, A828, A827, A831, A862, GOBOARD7:21;
suppose ( i19 = i1 & j1 = j29 ) ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A862; ::_thesis: verum
end;
suppose ( i19 = i1 & j29 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A862; ::_thesis: verum
end;
suppose ( i19 = i1 + 1 & j1 = j29 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A862; ::_thesis: verum
end;
suppose ( i19 = i1 + 1 & j29 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A862; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA863: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i19 = i2 & j19 = j1 ) or ( i19 = i2 & j19 + 1 = j1 ) or ( i19 = i2 + 1 & j19 = j1 ) or ( i19 = i2 + 1 & j19 + 1 = j1 ) ) by A835, A838, A840, A848, A846, A844, A850, A849, A819, A821, A823, A826, A830, A824, A831, A863, GOBOARD7:21;
suppose ( i19 = i2 & j19 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A863; ::_thesis: verum
end;
suppose ( i19 = i2 & j19 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A863; ::_thesis: verum
end;
suppose ( i19 = i2 + 1 & j19 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A863; ::_thesis: verum
end;
suppose ( i19 = i2 + 1 & j19 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A863; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA864: ( i1 = i2 + 1 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then A865: j1 = j19 by A835, A838, A840, A848, A846, A844, A850, A849, A819, A821, A823, A830, A828, A829, A831, GOBOARD7:20;
now__::_thesis:_contradiction
percases ( i2 = i19 or i2 = i19 + 1 or i2 + 1 = i19 ) by A835, A838, A840, A848, A846, A844, A850, A849, A819, A821, A823, A830, A828, A829, A831, A864, GOBOARD7:23;
suppose i2 = i19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A864, A865; ::_thesis: verum
end;
suppose i2 = i19 + 1 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A864, A865; ::_thesis: verum
end;
suppose i2 + 1 = i19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A864, A865; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA866: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then A867: j1 = j19 by A835, A838, A840, A848, A846, A844, A850, A849, A819, A821, A826, A830, A828, A825, A831, GOBOARD7:20;
now__::_thesis:_contradiction
percases ( i2 = i29 or i2 = i29 + 1 or i2 + 1 = i29 ) by A835, A838, A840, A848, A846, A844, A850, A849, A819, A821, A826, A830, A828, A825, A831, A866, GOBOARD7:23;
suppose i2 = i29 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A866, A867; ::_thesis: verum
end;
suppose i2 = i29 + 1 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A866, A867; ::_thesis: verum
end;
suppose i2 + 1 = i29 ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A866, A867; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA868: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i19 = i2 & j29 = j1 ) or ( i19 = i2 & j29 + 1 = j1 ) or ( i19 = i2 + 1 & j29 = j1 ) or ( i19 = i2 + 1 & j29 + 1 = j1 ) ) by A835, A838, A840, A848, A846, A844, A850, A849, A819, A821, A823, A826, A828, A827, A831, A868, GOBOARD7:21;
suppose ( i19 = i2 & j29 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A868; ::_thesis: verum
end;
suppose ( i19 = i2 & j29 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A868; ::_thesis: verum
end;
suppose ( i19 = i2 + 1 & j29 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A868; ::_thesis: verum
end;
suppose ( i19 = i2 + 1 & j29 + 1 = j1 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A868; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA869: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then A870: i1 = i19 by A835, A838, A840, A842, A848, A844, A845, A849, A819, A821, A823, A826, A830, A824, A831, GOBOARD7:19;
now__::_thesis:_contradiction
percases ( j2 = j19 or j2 = j19 + 1 or j2 + 1 = j19 ) by A835, A838, A840, A842, A848, A844, A845, A849, A819, A821, A823, A826, A830, A824, A831, A869, GOBOARD7:22;
suppose j2 = j19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A869, A870; ::_thesis: verum
end;
suppose j2 = j19 + 1 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A869, A870; ::_thesis: verum
end;
suppose j2 + 1 = j19 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A869, A870; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA871: ( i1 = i2 & j1 = j2 + 1 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i1 = i19 & j2 = j19 ) or ( i1 = i19 & j2 + 1 = j19 ) or ( i1 = i19 + 1 & j2 = j19 ) or ( i1 = i19 + 1 & j2 + 1 = j19 ) ) by A835, A838, A840, A842, A848, A844, A845, A849, A819, A821, A823, A830, A828, A829, A831, A871, GOBOARD7:21;
suppose ( i1 = i19 & j2 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A871; ::_thesis: verum
end;
suppose ( i1 = i19 & j2 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A871; ::_thesis: verum
end;
suppose ( i1 = i19 + 1 & j2 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A871; ::_thesis: verum
end;
suppose ( i1 = i19 + 1 & j2 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A871; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA872: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( i1 = i29 & j2 = j19 ) or ( i1 = i29 & j2 + 1 = j19 ) or ( i1 = i29 + 1 & j2 = j19 ) or ( i1 = i29 + 1 & j2 + 1 = j19 ) ) by A835, A838, A840, A842, A848, A844, A845, A849, A819, A821, A826, A830, A828, A825, A831, A872, GOBOARD7:21;
suppose ( i1 = i29 & j2 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A872; ::_thesis: verum
end;
suppose ( i1 = i29 & j2 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A872; ::_thesis: verum
end;
suppose ( i1 = i29 + 1 & j2 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A872; ::_thesis: verum
end;
suppose ( i1 = i29 + 1 & j2 + 1 = j19 ) ; ::_thesis: contradiction
hence  contradiction by A710, A752, A756, A754, A817, A836, A833, A838, A819, A872; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA873: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then A874: i1 = i19 by A835, A838, A840, A842, A848, A844, A845, A849, A819, A821, A823, A826, A828, A827, A831, GOBOARD7:19;
now__::_thesis:_contradiction
percases ( j2 = j29 or j2 = j29 + 1 or j2 + 1 = j29 ) by A835, A838, A840, A842, A848, A844, A845, A849, A819, A821, A823, A826, A828, A827, A831, A873, GOBOARD7:22;
suppose j2 = j29 ; ::_thesis: contradiction
hence  contradiction by A738, A757, A817, A832, A840, A821, A873, A874; ::_thesis: verum
end;
suppose j2 = j29 + 1 ; ::_thesis: contradiction
hence  contradiction by A710, A752, A755, A754, A834, A833, A840, A819, A873, A874; ::_thesis: verum
end;
suppose j2 + 1 = j29 ; ::_thesis: contradiction
hence  contradiction by A746, A758, A817, A838, A821, A873, A874; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
then reconsider g = g as non constant standard special_circular_sequence by A650, A737, A701, A704, A706, FINSEQ_6:def_1, JORDAN8:4;
reconsider Lg9 = (L~ g) ` as Subset of (TOP-REAL 2) ;
A875: C c= Lg9
proof
let c be set ; :: according to TARSKI:def_3 ::_thesis: ( not c in C or c in Lg9 )
assume that
A876: c in C and
A877: not c in Lg9 ; ::_thesis: contradiction
reconsider c = c as Point of (TOP-REAL 2) by A876;
consider i being Element of NAT  such that
A878: 1 <= i and
A879: i + 1 <= len g and
A880: c in LSeg ((g /. i),(g /. (i + 1))) by A877, SPPOL_2:14, SUBSET_1:29;
A881: 1 <= i + (m -' 1) by A878, NAT_1:12;
 i + 1 in dom g by A878, A879, SEQ_4:134;
then A882: g /. (i + 1) = f /. ((i + 1) + (m -' 1)) by FINSEQ_5:27;
 (i + 1) + (m -' 1) = (i + (m -' 1)) + 1 ;
then A883: (i + (m -' 1)) + 1 <= (len g) + (m -' 1) by A879, XREAL_1:6;
 i in dom g by A878, A879, SEQ_4:134;
then  g /. i = f /. (i + (m -' 1)) by FINSEQ_5:27;
then  c in LSeg (f,(i + (m -' 1))) by A700, A880, A882, A881, A883, TOPREAL1:def_3;
then  c in (right_cell (f,(i + (m -' 1)),(Gauge (C,n)))) /\ (left_cell (f,(i + (m -' 1)),(Gauge (C,n)))) by A439, A700, A881, A883, GOBRD13:29;
then  c in right_cell (f,(i + (m -' 1)),(Gauge (C,n))) by XBOOLE_0:def_4;
then  right_cell (f,(i + (m -' 1)),(Gauge (C,n))) meets C by A876, XBOOLE_0:3;
hence  contradiction by A439, A700, A881, A883; ::_thesis: verum
end;
A884: LeftComp g is_a_component_of (L~ g) ` by GOBOARD9:def_1;
 (L~ g) ` is open by TOPS_1:3;
then A885: (L~ g) ` = Int ((L~ g) `) by TOPS_1:23;
A886: C meets LeftComp g
proof
 left_cell (f,m,(Gauge (C,n))) meets C by A439, A651, A658;
then consider p being set  such that
A887: p in left_cell (f,m,(Gauge (C,n))) and
A888: p in C by XBOOLE_0:3;
reconsider p = p as Element of (TOP-REAL 2) by A887;
now__::_thesis:_ex_p_being_Element_of_(TOP-REAL_2)_st_
(_p_in_C_&_p_in_LeftComp_g_)
reconsider u = p as Element of (Euclid 2) by TOPREAL3:8;
take p = p; ::_thesis: ( p in C & p in LeftComp g )
thus  p in C by A888; ::_thesis: p in LeftComp g
A889: Int (left_cell (g,1)) c= LeftComp g by A703, GOBOARD9:21;
 Int (left_cell (g,1,(Gauge (C,n)))) c= Int (left_cell (g,1)) by A704, A703, GOBRD13:33, TOPS_1:19;
then  Int (left_cell (g,1,(Gauge (C,n)))) c= LeftComp g by A889, XBOOLE_1:1;
then  Int (left_cell (f,((m -' 1) + 1),(Gauge (C,n)))) c= LeftComp g by A652, A699, A703, GOBRD13:32;
then A890: Int (left_cell (f,m,(Gauge (C,n)))) c= LeftComp g by A651, XREAL_1:235;
consider r being real number  such that
A891: r > 0 and
A892: Ball (u,r) c= (L~ g) ` by A875, A885, A888, GOBOARD6:5;
reconsider r = r as Real by XREAL_0:def_1;
reconsider B = Ball (u,r) as non empty Subset of (TOP-REAL 2) by A4, A891, TBSP_1:11, TOPMETR:12;
A893: B is open by GOBOARD6:3;
A894: left_cell (f,m,(Gauge (C,n))) = Cl (Int (left_cell (f,m,(Gauge (C,n))))) by A651, A658, A679;
 p in Ball (u,r) by A891, TBSP_1:11;
then A895: Int (left_cell (f,m,(Gauge (C,n)))) meets B by A887, A894, A893, TOPS_1:12;
A896: p in B by A891, TBSP_1:11;
 B is connected by SPRECT_3:7;
then  B c= LeftComp g by A884, A892, A890, A895, GOBOARD9:4;
hence  p in LeftComp g by A896; ::_thesis: verum
end;
hence  C meets LeftComp g by XBOOLE_0:3; ::_thesis: verum
end;
A897: L~ g c= L~ f by JORDAN3:40;
A898: RightComp g is_a_component_of (L~ g) ` by GOBOARD9:def_2;
 m = 1
proof
A899: for n being Element of NAT st 1 <= n holds
(n -' 1) + 2 = n + 1
proof
let n be Element of NAT ; ::_thesis: ( 1 <= n implies (n -' 1) + 2 = n + 1 )
assume  1 <= n ; ::_thesis: (n -' 1) + 2 = n + 1
hence (n -' 1) + 2 = (n + 2) -' 1 by NAT_D:38
.= ((n + 1) + 1) - 1 by NAT_D:37
.= n + 1 ;
::_thesis: verum
end;
assume  m <> 1 ; ::_thesis: contradiction
then A900: 1 < m by A651, XXREAL_0:1;
A901: for n being Element of NAT st 1 <= n & n <= m -' 1 holds
not f /. n in L~ g
proof
A902: 2 <= len (Gauge (C,n)) by A2, NAT_1:12;
let n be Element of NAT ; ::_thesis: ( 1 <= n & n <= m -' 1 implies not f /. n in L~ g )
assume that
A903: 1 <= n and
A904: n <= m -' 1 ; ::_thesis: not f /. n in L~ g
set p = f /. n;
A905: n <= len f by A699, A904, XXREAL_0:2;
then A906: f /. n in Values (Gauge (C,n)) by A439, A903, JORDAN9:6;
assume  f /. n in L~ g ; ::_thesis: contradiction
then consider j being Element of NAT  such that
A907: (m -' 1) + 1 <= j and
A908: j + 1 <= len f and
A909: f /. n in LSeg (f,j) by A699, JORDAN9:7;
A910: j + 1 <= k by A176, A908;
A911: j < k by A709, A908, NAT_1:13;
A912: n < (m -' 1) + 1 by A904, NAT_1:13;
then A913: n < j by A907, XXREAL_0:2;
A914: (m -' 1) + 1 = m by A651, XREAL_1:235;
then A915: 1 < j by A900, A907, XXREAL_0:2;
percases ( f /. n = f /. j or f /. n = f /. (j + 1) ) by A6, A439, A908, A909, A915, A902, A906, JORDAN9:23;
supposeA916: f /. n = f /. j ; ::_thesis: contradiction
A917: n <> len (F . j) by A176, A907, A912;
 n <= len (F . j) by A176, A913;
then A918: n in dom (F . j) by A903, FINSEQ_3:25;
(F . j) /. n = (F . n) /. n by A565, A903, A913
.= f /. n by A709, A565, A903, A905
.= (F . j) /. j by A565, A915, A911, A916
.= (F . j) /. (len (F . j)) by A176 ;
hence  contradiction by A647, A915, A911, A918, A917; ::_thesis: verum
end;
supposeA919: f /. n = f /. (j + 1) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( j + 1 = k or j + 1 < k ) by A910, XXREAL_0:1;
supposeA920: j + 1 = k ; ::_thesis: contradiction
A921: n <> len (F . m) by A176, A912, A914;
 n <= len (F . m) by A176, A912, A914;
then A922: n in dom (F . m) by A903, FINSEQ_3:25;
(F . m) /. n = (F . n) /. n by A565, A903, A912, A914
.= f /. n by A709, A565, A903, A905
.= (F . m) /. m by A650, A709, A651, A653, A565, A919, A920
.= (F . m) /. (len (F . m)) by A176 ;
hence  contradiction by A647, A709, A651, A654, A922, A921; ::_thesis: verum
end;
supposeA923: j + 1 < k ; ::_thesis: contradiction
set l = j + 1;
A924: 1 <= j + 1 by NAT_1:11;
A925: n < n + 1 by XREAL_1:29;
A926: n + 1 < j + 1 by A913, XREAL_1:6;
then A927: n <> len (F . (j + 1)) by A176, A925;
A928: n < j + 1 by A925, A926, XXREAL_0:2;
then  n <= len (F . (j + 1)) by A176;
then A929: n in dom (F . (j + 1)) by A903, FINSEQ_3:25;
(F . (j + 1)) /. n = (F . n) /. n by A565, A903, A928
.= f /. n by A709, A565, A903, A905
.= (F . (j + 1)) /. (j + 1) by A565, A919, A923, A924
.= (F . (j + 1)) /. (len (F . (j + 1))) by A176 ;
hence  contradiction by A647, A923, A929, A927, NAT_1:11; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
 C meets LeftComp (Rev g)
proof
 1 <= len g by A703, XREAL_1:145;
then A930: ((len g) -' 1) + 2 = (len g) + 1 by A899;
A931: 1 - 1 < m - 1 by A900, XREAL_1:9;
A932: (m -' 1) + 2 = m + 1 by A651, A899;
set l = (m -' 1) + ((len g) -' 1);
set a = f /. (m -' 1);
set rg = Rev g;
set p = (Rev g) /. 1;
set q = (Rev g) /. 2;
A933: (1 + 1) - 1 <= (len g) - 1 by A702, XREAL_1:9;
 (1 + 1) -' 1 <= (len g) -' 1 by A702, NAT_D:42;
then A934: 1 <= (len g) -' 1 by NAT_D:34;
then  (m -' 1) + 1 <= (m -' 1) + ((len g) -' 1) by XREAL_1:6;
then  m -' 1 < (m -' 1) + ((len g) -' 1) by NAT_1:13;
then A935: m -' 1 <> len (F . ((m -' 1) + ((len g) -' 1))) by A176;
A936: 1 + 1 <= len (Rev g) by A702, FINSEQ_5:def_3;
then  (1 + 1) -' 1 <= (len (Rev g)) -' 1 by NAT_D:42;
then A937: 1 <= (len (Rev g)) -' 1 by NAT_D:34;
A938: Rev g is_sequence_on Gauge (C,n) by A704, JORDAN9:5;
then consider p1, p2, q1, q2 being Element of NAT  such that
A939: [p1,p2] in Indices (Gauge (C,n)) and
A940: (Rev g) /. 1 = (Gauge (C,n)) * (p1,p2) and
A941: [q1,q2] in Indices (Gauge (C,n)) and
A942: (Rev g) /. 2 = (Gauge (C,n)) * (q1,q2) and
A943: ( ( p1 = q1 & p2 + 1 = q2 ) or ( p1 + 1 = q1 & p2 = q2 ) or ( p1 = q1 + 1 & p2 = q2 ) or ( p1 = q1 & p2 = q2 + 1 ) ) by A936, JORDAN8:3;
A944: 1 <= p1 by A939, MATRIX_1:38;
A945: p2 <= width (Gauge (C,n)) by A939, MATRIX_1:38;
A946: p1 <= len (Gauge (C,n)) by A939, MATRIX_1:38;
A947: 1 <= p2 by A939, MATRIX_1:38;
A948: (Rev g) /. 1 = f /. m by A650, A701, FINSEQ_5:65;
 (len g) -' 1 <= len g by NAT_D:44;
then A949: (len g) -' 1 in dom g by A934, FINSEQ_3:25;
then A950: (Rev g) /. 2 = g /. ((len g) -' 1) by A930, FINSEQ_5:66
.= f /. ((m -' 1) + ((len g) -' 1)) by A949, FINSEQ_5:27 ;
 1 < len (Rev g) by A936, NAT_1:13;
then A951: ((len (Rev g)) -' 1) + 1 = len (Rev g) by XREAL_1:235;
A952: (m -' 1) + ((len g) -' 1) = (m + ((len g) -' 1)) -' 1 by A651, NAT_D:38
.= (((len g) -' 1) + m) - 1 by A934, NAT_D:37
.= (((len g) - 1) + m) - 1 by A933, XREAL_0:def_2
.= (((k - (m - 1)) - 1) + m) - 1 by A709, A700, A931, XREAL_0:def_2
.= k - 1 ;
then A953: (Rev g) /. 1 = f /. (((m -' 1) + ((len g) -' 1)) + 1) by A709, A701, FINSEQ_5:65;
A954: (m -' 1) + 1 = m by A651, XREAL_1:235;
then A955: 1 <= m -' 1 by A900, NAT_1:13;
then A956: left_cell (f,(m -' 1),(Gauge (C,n))) meets C by A439, A653, A954;
 m -' 1 <= (m -' 1) + ((len g) -' 1) by NAT_1:11;
then  m -' 1 <= len (F . ((m -' 1) + ((len g) -' 1))) by A176;
then A957: m -' 1 in dom (F . ((m -' 1) + ((len g) -' 1))) by A955, FINSEQ_3:25;
 not f /. (m -' 1) in L~ g by A901, A955;
then A958: not f /. (m -' 1) in L~ (Rev g) by SPPOL_2:22;
A959: k = ((m -' 1) + ((len g) -' 1)) + 1 by A952;
then A960: (m -' 1) + ((len g) -' 1) < k by XREAL_1:29;
 (len g) -' 1 <= (m -' 1) + ((len g) -' 1) by NAT_1:11;
then A961: 1 <= (m -' 1) + ((len g) -' 1) by A934, XXREAL_0:2;
then A962: left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) meets C by A439, A709, A959;
percases ( ( p1 = q1 & p2 + 1 = q2 ) or ( p1 + 1 = q1 & p2 = q2 ) or ( p1 = q1 + 1 & p2 = q2 ) or ( p1 = q1 & p2 = q2 + 1 ) ) by A943;
supposeA963: ( p1 = q1 & p2 + 1 = q2 ) ; ::_thesis: C meets LeftComp (Rev g)
consider a1, a2, p91, p92 being Element of NAT  such that
A964: [a1,a2] in Indices (Gauge (C,n)) and
A965: f /. (m -' 1) = (Gauge (C,n)) * (a1,a2) and
A966: [p91,p92] in Indices (Gauge (C,n)) and
A967: (Rev g) /. 1 = (Gauge (C,n)) * (p91,p92) and
A968: ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A652, A653, A948, A954, A955, JORDAN8:3;
A969: 1 <= a2 by A964, MATRIX_1:38;
thus  C meets LeftComp (Rev g) ::_thesis: verum
proof
percases ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A968;
supposeA970: ( a1 = p91 & a2 + 1 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
A971: m -' 1 <= m by A954, NAT_1:11;
A972: f /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A709, A699, A565, A955
.= (F . m) /. (m -' 1) by A565, A955, A971 ;
A973: 2 in dom g by A702, FINSEQ_3:25;
 ((len (Rev g)) -' 1) + 2 = (len g) + 1 by A930, FINSEQ_5:def_3;
then A974: (Rev g) /. ((len (Rev g)) -' 1) = g /. 2 by A973, FINSEQ_5:66
.= f /. (m + 1) by A932, A973, FINSEQ_5:27 ;
A975: L~ (Rev g) c= L~ f by A897, SPPOL_2:22;
A976: (Rev g) /. 1 = g /. 1 by A650, A737, A701, FINSEQ_5:65
.= (Rev g) /. (len g) by FINSEQ_5:65
.= (Rev g) /. (len (Rev g)) by FINSEQ_5:def_3 ;
A977: (F . k) | (m + 1) = F . (m + 1) by A564, A709, A658;
A978: a1 = p1 by A939, A940, A966, A967, A970, GOBOARD1:5;
A979: f /. ((m -' 1) + 1) = (F . m) /. m by A709, A651, A653, A565, A954;
A980: (m -' 1) + 1 <= len (F . m) by A176, A954;
set rc = (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g));
A981: a2 + 1 > a2 by NAT_1:13;
A982: a2 + 1 = p2 by A939, A940, A966, A967, A970, GOBOARD1:5;
then A983: p2 -' 1 = a2 by NAT_D:34;
left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,p2) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A963, GOBRD13:27
.= front_right_cell ((F . m),(m -' 1),(Gauge (C,n))) by A439, A948, A954, A955, A939, A940, A964, A965, A978, A982, A980, A972, A979, GOBRD13:35 ;
then  F . (m + 1) turns_right m -' 1, Gauge (C,n) by A513, A900, A962;
then A984: f turns_right m -' 1, Gauge (C,n) by A955, A932, A977, GOBRD13:43;
A985: p2 + 1 > a2 + 1 by A982, NAT_1:13;
then A986: [(p1 + 1),p2] in Indices (Gauge (C,n)) by A948, A954, A939, A940, A964, A965, A981, A984, GOBRD13:def_6;
then A987: p1 + 1 <= len (Gauge (C,n)) by MATRIX_1:38;
 f /. (m + 1) = (Gauge (C,n)) * ((p1 + 1),p2) by A948, A954, A932, A939, A940, A964, A965, A985, A981, A984, GOBRD13:def_6;
then  left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,a2) by A938, A937, A951, A939, A940, A986, A983, A974, A976, GOBRD13:25;
then  f /. (m -' 1) in left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) by A944, A945, A965, A969, A978, A982, A987, JORDAN9:20;
then A988: f /. (m -' 1) in (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) by A958, XBOOLE_0:def_5;
A989: LeftComp (Rev g) is_a_component_of (L~ (Rev g)) ` by GOBOARD9:def_1;
 (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) c= LeftComp (Rev g) by A938, A937, A951, JORDAN9:27;
hence  C meets LeftComp (Rev g) by A653, A659, A954, A955, A958, A988, A975, A989; ::_thesis: verum
end;
supposeA990: ( a1 + 1 = p91 & a2 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
then  a1 + 1 = p1 by A939, A940, A966, A967, GOBOARD1:5;
then A991: q1 -' 1 = a1 by A963, NAT_D:34;
 a2 = p2 by A939, A940, A966, A967, A990, GOBOARD1:5;
then  right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),a1,a2) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A963, A991, GOBRD13:28
.= left_cell (f,(m -' 1),(Gauge (C,n))) by A439, A653, A948, A954, A955, A964, A965, A966, A967, A990, GOBRD13:23 ;
hence  C meets LeftComp (Rev g) by A439, A709, A959, A961, A956; ::_thesis: verum
end;
supposeA992: ( a1 = p91 + 1 & a2 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
then A993: a2 = p2 by A939, A940, A966, A967, GOBOARD1:5;
 a1 = p1 + 1 by A939, A940, A966, A967, A992, GOBOARD1:5;
then  right_cell (f,(m -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,p2) by A650, A652, A653, A701, A954, A955, A939, A940, A964, A965, A993, FINSEQ_5:65, GOBRD13:26
.= left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A963, GOBRD13:27 ;
hence  C meets LeftComp (Rev g) by A439, A653, A954, A955, A962; ::_thesis: verum
end;
supposeA994: ( a1 = p91 & a2 = p92 + 1 ) ; ::_thesis: C meets LeftComp (Rev g)
then A995: a2 = q2 by A939, A940, A963, A966, A967, GOBOARD1:5;
A996: a1 = q1 by A939, A940, A963, A966, A967, A994, GOBOARD1:5;
(F . ((m -' 1) + ((len g) -' 1))) /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A565, A955, NAT_1:11
.= (Rev g) /. 2 by A709, A699, A565, A955, A942, A965, A996, A995
.= (F . ((m -' 1) + ((len g) -' 1))) /. ((m -' 1) + ((len g) -' 1)) by A565, A960, A961, A950
.= (F . ((m -' 1) + ((len g) -' 1))) /. (len (F . ((m -' 1) + ((len g) -' 1)))) by A176 ;
hence  C meets LeftComp (Rev g) by A647, A960, A961, A957, A935; ::_thesis: verum
end;
end;
end;
end;
supposeA997: ( p1 + 1 = q1 & p2 = q2 ) ; ::_thesis: C meets LeftComp (Rev g)
consider a1, a2, p91, p92 being Element of NAT  such that
A998: [a1,a2] in Indices (Gauge (C,n)) and
A999: f /. (m -' 1) = (Gauge (C,n)) * (a1,a2) and
A1000: [p91,p92] in Indices (Gauge (C,n)) and
A1001: (Rev g) /. 1 = (Gauge (C,n)) * (p91,p92) and
A1002: ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A652, A653, A948, A954, A955, JORDAN8:3;
A1003: 1 <= a2 by A998, MATRIX_1:38;
A1004: a2 <= width (Gauge (C,n)) by A998, MATRIX_1:38;
A1005: 1 <= a1 by A998, MATRIX_1:38;
thus  C meets LeftComp (Rev g) ::_thesis: verum
proof
percases ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A1002;
supposeA1006: ( a1 = p91 & a2 + 1 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
then  a2 + 1 = p2 by A939, A940, A1000, A1001, GOBOARD1:5;
then A1007: q2 -' 1 = a2 by A997, NAT_D:34;
A1008: a1 = p1 by A939, A940, A1000, A1001, A1006, GOBOARD1:5;
right_cell (f,(m -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),a1,a2) by A439, A653, A948, A954, A955, A998, A999, A1000, A1001, A1006, GOBRD13:22
.= left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A997, A1008, A1007, GOBRD13:25 ;
hence  C meets LeftComp (Rev g) by A439, A653, A954, A955, A962; ::_thesis: verum
end;
supposeA1009: ( a1 + 1 = p91 & a2 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
A1010: m -' 1 <= m by A954, NAT_1:11;
A1011: f /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A709, A699, A565, A955
.= (F . m) /. (m -' 1) by A565, A955, A1010 ;
A1012: 2 in dom g by A702, FINSEQ_3:25;
 ((len (Rev g)) -' 1) + 2 = (len g) + 1 by A930, FINSEQ_5:def_3;
then A1013: (Rev g) /. ((len (Rev g)) -' 1) = g /. 2 by A1012, FINSEQ_5:66
.= f /. (m + 1) by A932, A1012, FINSEQ_5:27 ;
A1014: L~ (Rev g) c= L~ f by A897, SPPOL_2:22;
A1015: (F . k) | (m + 1) = F . (m + 1) by A564, A709, A658;
A1016: (m -' 1) + 1 <= len (F . m) by A176, A954;
A1017: a2 = p2 by A939, A940, A1000, A1001, A1009, GOBOARD1:5;
A1018: (Rev g) /. 1 = g /. 1 by A650, A737, A701, FINSEQ_5:65
.= (Rev g) /. (len g) by FINSEQ_5:65
.= (Rev g) /. (len (Rev g)) by FINSEQ_5:def_3 ;
set rc = (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g));
A1019: p1 < p1 + 1 by XREAL_1:29;
A1020: f /. ((m -' 1) + 1) = (F . m) /. m by A709, A651, A653, A565, A954;
A1021: (a2 -' 1) + 1 = a2 by A1003, XREAL_1:235;
A1022: a1 + 1 = p1 by A939, A940, A1000, A1001, A1009, GOBOARD1:5;
then A1023: a1 = p1 -' 1 by NAT_D:34;
left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,(p2 -' 1)) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A997, GOBRD13:25
.= front_right_cell ((F . m),(m -' 1),(Gauge (C,n))) by A439, A948, A954, A955, A939, A940, A998, A999, A1022, A1017, A1016, A1011, A1020, GOBRD13:37 ;
then  F . (m + 1) turns_right m -' 1, Gauge (C,n) by A513, A900, A962;
then A1024: f turns_right m -' 1, Gauge (C,n) by A955, A932, A1015, GOBRD13:43;
A1025: a1 < a1 + 1 by XREAL_1:29;
then A1026: [p1,(p2 -' 1)] in Indices (Gauge (C,n)) by A948, A954, A939, A940, A998, A999, A1022, A1019, A1024, GOBRD13:def_6;
then A1027: 1 <= a2 -' 1 by A1017, MATRIX_1:38;
 f /. (m + 1) = (Gauge (C,n)) * (p1,(p2 -' 1)) by A948, A954, A932, A939, A940, A998, A999, A1022, A1025, A1019, A1024, GOBRD13:def_6;
then  left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),a1,(a2 -' 1)) by A938, A937, A951, A939, A940, A1017, A1026, A1023, A1013, A1021, A1018, GOBRD13:21;
then  f /. (m -' 1) in left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) by A946, A999, A1005, A1004, A1022, A1021, A1027, JORDAN9:20;
then A1028: f /. (m -' 1) in (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) by A958, XBOOLE_0:def_5;
A1029: LeftComp (Rev g) is_a_component_of (L~ (Rev g)) ` by GOBOARD9:def_1;
 (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) c= LeftComp (Rev g) by A938, A937, A951, JORDAN9:27;
hence  C meets LeftComp (Rev g) by A653, A659, A954, A955, A958, A1028, A1014, A1029; ::_thesis: verum
end;
supposeA1030: ( a1 = p91 + 1 & a2 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
then A1031: a2 = q2 by A939, A940, A997, A1000, A1001, GOBOARD1:5;
A1032: a1 = q1 by A939, A940, A997, A1000, A1001, A1030, GOBOARD1:5;
(F . ((m -' 1) + ((len g) -' 1))) /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A565, A955, NAT_1:11
.= (Rev g) /. 2 by A709, A699, A565, A955, A942, A999, A1032, A1031
.= (F . ((m -' 1) + ((len g) -' 1))) /. ((m -' 1) + ((len g) -' 1)) by A565, A960, A961, A950
.= (F . ((m -' 1) + ((len g) -' 1))) /. (len (F . ((m -' 1) + ((len g) -' 1)))) by A176 ;
hence  C meets LeftComp (Rev g) by A647, A960, A961, A957, A935; ::_thesis: verum
end;
supposeA1033: ( a1 = p91 & a2 = p92 + 1 ) ; ::_thesis: C meets LeftComp (Rev g)
then A1034: a2 = p2 + 1 by A939, A940, A1000, A1001, GOBOARD1:5;
A1035: a1 = p1 by A939, A940, A1000, A1001, A1033, GOBOARD1:5;
right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,p2) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A997, GOBRD13:26
.= left_cell (f,(m -' 1),(Gauge (C,n))) by A650, A652, A653, A701, A954, A955, A939, A940, A998, A999, A1035, A1034, FINSEQ_5:65, GOBRD13:27 ;
hence  C meets LeftComp (Rev g) by A439, A709, A959, A961, A956; ::_thesis: verum
end;
end;
end;
end;
supposeA1036: ( p1 = q1 + 1 & p2 = q2 ) ; ::_thesis: C meets LeftComp (Rev g)
consider a1, a2, p91, p92 being Element of NAT  such that
A1037: [a1,a2] in Indices (Gauge (C,n)) and
A1038: f /. (m -' 1) = (Gauge (C,n)) * (a1,a2) and
A1039: [p91,p92] in Indices (Gauge (C,n)) and
A1040: (Rev g) /. 1 = (Gauge (C,n)) * (p91,p92) and
A1041: ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A652, A653, A948, A954, A955, JORDAN8:3;
A1042: a1 <= len (Gauge (C,n)) by A1037, MATRIX_1:38;
thus  C meets LeftComp (Rev g) ::_thesis: verum
proof
percases ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A1041;
supposeA1043: ( a1 = p91 & a2 + 1 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
then  a2 + 1 = p2 by A939, A940, A1039, A1040, GOBOARD1:5;
then A1044: q2 -' 1 = a2 by A1036, NAT_D:34;
 a1 = p1 by A939, A940, A1039, A1040, A1043, GOBOARD1:5;
then A1045: q1 = a1 -' 1 by A1036, NAT_D:34;
right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),q1,(q2 -' 1)) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A1036, GOBRD13:24
.= left_cell (f,(m -' 1),(Gauge (C,n))) by A439, A653, A948, A954, A955, A1037, A1038, A1039, A1040, A1043, A1045, A1044, GOBRD13:21 ;
hence  C meets LeftComp (Rev g) by A439, A709, A959, A961, A956; ::_thesis: verum
end;
supposeA1046: ( a1 + 1 = p91 & a2 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
then A1047: a2 = p2 by A939, A940, A1039, A1040, GOBOARD1:5;
A1048: a1 + 1 = p1 by A939, A940, A1039, A1040, A1046, GOBOARD1:5;
(F . ((m -' 1) + ((len g) -' 1))) /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A565, A955, NAT_1:11
.= (Rev g) /. 2 by A709, A699, A565, A955, A942, A1036, A1038, A1048, A1047
.= (F . ((m -' 1) + ((len g) -' 1))) /. ((m -' 1) + ((len g) -' 1)) by A565, A960, A961, A950
.= (F . ((m -' 1) + ((len g) -' 1))) /. (len (F . ((m -' 1) + ((len g) -' 1)))) by A176 ;
hence  C meets LeftComp (Rev g) by A647, A960, A961, A957, A935; ::_thesis: verum
end;
supposeA1049: ( a1 = p91 + 1 & a2 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
A1050: m -' 1 <= m by A954, NAT_1:11;
A1051: f /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A709, A699, A565, A955
.= (F . m) /. (m -' 1) by A565, A955, A1050 ;
A1052: 2 in dom g by A702, FINSEQ_3:25;
 ((len (Rev g)) -' 1) + 2 = (len g) + 1 by A930, FINSEQ_5:def_3;
then A1053: (Rev g) /. ((len (Rev g)) -' 1) = g /. 2 by A1052, FINSEQ_5:66
.= f /. (m + 1) by A932, A1052, FINSEQ_5:27 ;
A1054: L~ (Rev g) c= L~ f by A897, SPPOL_2:22;
set rc = (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g));
A1055: LeftComp (Rev g) is_a_component_of (L~ (Rev g)) ` by GOBOARD9:def_1;
A1056: p1 -' 1 = q1 by A1036, NAT_D:34;
A1057: (F . k) | (m + 1) = F . (m + 1) by A564, A709, A658;
A1058: a1 = p1 + 1 by A939, A940, A1039, A1040, A1049, GOBOARD1:5;
A1059: f /. ((m -' 1) + 1) = (F . m) /. m by A709, A651, A653, A565, A954;
A1060: (m -' 1) + 1 <= len (F . m) by A176, A954;
A1061: a2 = p2 by A939, A940, A1039, A1040, A1049, GOBOARD1:5;
left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),q1,q2) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A1036, GOBRD13:23
.= front_right_cell ((F . m),(m -' 1),(Gauge (C,n))) by A439, A948, A954, A955, A939, A940, A1036, A1037, A1038, A1058, A1061, A1056, A1060, A1051, A1059, GOBRD13:39 ;
then  F . (m + 1) turns_right m -' 1, Gauge (C,n) by A513, A900, A962;
then A1062: f turns_right m -' 1, Gauge (C,n) by A955, A932, A1057, GOBRD13:43;
 p1 + 1 > p1 by XREAL_1:29;
then A1063: a1 + 1 > p1 by A1058, NAT_1:13;
then A1064: [p1,(p2 + 1)] in Indices (Gauge (C,n)) by A948, A954, A939, A940, A1037, A1038, A1061, A1062, GOBRD13:def_6;
then A1065: p2 + 1 <= width (Gauge (C,n)) by MATRIX_1:38;
 a2 + 1 > p2 by A1061, NAT_1:13;
then A1066: f /. (m + 1) = (Gauge (C,n)) * (p1,(p2 + 1)) by A948, A954, A932, A939, A940, A1037, A1038, A1061, A1063, A1062, GOBRD13:def_6;
(Rev g) /. 1 = g /. 1 by A650, A737, A701, FINSEQ_5:65
.= (Rev g) /. (len g) by FINSEQ_5:65
.= (Rev g) /. (len (Rev g)) by FINSEQ_5:def_3 ;
then  left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,p2) by A938, A937, A951, A939, A940, A1066, A1064, A1053, GOBRD13:27;
then  f /. (m -' 1) in left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) by A944, A947, A1038, A1042, A1058, A1061, A1065, JORDAN9:20;
then A1067: f /. (m -' 1) in (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) by A958, XBOOLE_0:def_5;
 (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) c= LeftComp (Rev g) by A938, A937, A951, JORDAN9:27;
hence  C meets LeftComp (Rev g) by A653, A659, A954, A955, A958, A1067, A1054, A1055; ::_thesis: verum
end;
supposeA1068: ( a1 = p91 & a2 = p92 + 1 ) ; ::_thesis: C meets LeftComp (Rev g)
then  a1 = p1 by A939, A940, A1039, A1040, GOBOARD1:5;
then A1069: q1 = a1 -' 1 by A1036, NAT_D:34;
 a2 = p2 + 1 by A939, A940, A1039, A1040, A1068, GOBOARD1:5;
then  right_cell (f,(m -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),q1,q2) by A650, A652, A653, A701, A954, A955, A1036, A1037, A1038, A1039, A1040, A1068, A1069, FINSEQ_5:65, GOBRD13:28
.= left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A1036, GOBRD13:23 ;
hence  C meets LeftComp (Rev g) by A439, A653, A954, A955, A962; ::_thesis: verum
end;
end;
end;
end;
supposeA1070: ( p1 = q1 & p2 = q2 + 1 ) ; ::_thesis: C meets LeftComp (Rev g)
consider a1, a2, p91, p92 being Element of NAT  such that
A1071: [a1,a2] in Indices (Gauge (C,n)) and
A1072: f /. (m -' 1) = (Gauge (C,n)) * (a1,a2) and
A1073: [p91,p92] in Indices (Gauge (C,n)) and
A1074: (Rev g) /. 1 = (Gauge (C,n)) * (p91,p92) and
A1075: ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A652, A653, A948, A954, A955, JORDAN8:3;
A1076: a2 <= width (Gauge (C,n)) by A1071, MATRIX_1:38;
thus  C meets LeftComp (Rev g) ::_thesis: verum
proof
percases ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A1075;
supposeA1077: ( a1 = p91 & a2 + 1 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
then A1078: a2 + 1 = p2 by A939, A940, A1073, A1074, GOBOARD1:5;
A1079: a1 = p1 by A939, A940, A1073, A1074, A1077, GOBOARD1:5;
(F . ((m -' 1) + ((len g) -' 1))) /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A565, A955, NAT_1:11
.= (Rev g) /. 2 by A709, A699, A565, A955, A942, A1070, A1072, A1079, A1078
.= (F . ((m -' 1) + ((len g) -' 1))) /. ((m -' 1) + ((len g) -' 1)) by A565, A960, A961, A950
.= (F . ((m -' 1) + ((len g) -' 1))) /. (len (F . ((m -' 1) + ((len g) -' 1)))) by A176 ;
hence  C meets LeftComp (Rev g) by A647, A960, A961, A957, A935; ::_thesis: verum
end;
supposeA1080: ( a1 + 1 = p91 & a2 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
then  a2 = p2 by A939, A940, A1073, A1074, GOBOARD1:5;
then A1081: a2 -' 1 = q2 by A1070, NAT_D:34;
 a1 + 1 = p1 by A939, A940, A1073, A1074, A1080, GOBOARD1:5;
then A1082: a1 = q1 -' 1 by A1070, NAT_D:34;
right_cell (f,(m -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),a1,(a2 -' 1)) by A439, A653, A948, A954, A955, A1071, A1072, A1073, A1074, A1080, GOBRD13:24
.= left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A1070, A1082, A1081, GOBRD13:21 ;
hence  C meets LeftComp (Rev g) by A439, A653, A954, A955, A962; ::_thesis: verum
end;
supposeA1083: ( a1 = p91 + 1 & a2 = p92 ) ; ::_thesis: C meets LeftComp (Rev g)
then  a2 = p2 by A939, A940, A1073, A1074, GOBOARD1:5;
then A1084: a2 -' 1 = q2 by A1070, NAT_D:34;
A1085: a1 = p1 + 1 by A939, A940, A1073, A1074, A1083, GOBOARD1:5;
right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),q1,q2) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A1070, GOBRD13:22
.= left_cell (f,(m -' 1),(Gauge (C,n))) by A650, A652, A653, A701, A954, A955, A1070, A1071, A1072, A1073, A1074, A1083, A1085, A1084, FINSEQ_5:65, GOBRD13:25 ;
hence  C meets LeftComp (Rev g) by A439, A709, A959, A961, A956; ::_thesis: verum
end;
supposeA1086: ( a1 = p91 & a2 = p92 + 1 ) ; ::_thesis: C meets LeftComp (Rev g)
then A1087: a2 = p2 + 1 by A939, A940, A1073, A1074, GOBOARD1:5;
A1088: f /. ((m -' 1) + 1) = (F . m) /. m by A709, A651, A653, A565, A954;
A1089: 2 in dom g by A702, FINSEQ_3:25;
 ((len (Rev g)) -' 1) + 2 = (len g) + 1 by A930, FINSEQ_5:def_3;
then A1090: (Rev g) /. ((len (Rev g)) -' 1) = g /. 2 by A1089, FINSEQ_5:66
.= f /. (m + 1) by A932, A1089, FINSEQ_5:27 ;
A1091: (p1 -' 1) + 1 = p1 by A944, XREAL_1:235;
A1092: m -' 1 <= m by A954, NAT_1:11;
A1093: f /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A709, A699, A565, A955
.= (F . m) /. (m -' 1) by A565, A955, A1092 ;
A1094: p2 -' 1 = q2 by A1070, NAT_D:34;
set rc = (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g));
A1095: p2 + 1 > p2 by NAT_1:13;
A1096: (Rev g) /. 1 = g /. 1 by A650, A737, A701, FINSEQ_5:65
.= (Rev g) /. (len g) by FINSEQ_5:65
.= (Rev g) /. (len (Rev g)) by FINSEQ_5:def_3 ;
A1097: (m -' 1) + 1 <= len (F . m) by A176, A954;
A1098: (F . k) | (m + 1) = F . (m + 1) by A564, A709, A658;
A1099: L~ (Rev g) c= L~ f by A897, SPPOL_2:22;
A1100: a1 = p1 by A939, A940, A1073, A1074, A1086, GOBOARD1:5;
left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),(q1 -' 1),q2) by A439, A709, A952, A961, A950, A953, A939, A940, A941, A942, A1070, GOBRD13:21
.= front_right_cell ((F . m),(m -' 1),(Gauge (C,n))) by A439, A948, A954, A955, A939, A940, A1070, A1071, A1072, A1100, A1087, A1094, A1097, A1093, A1088, GOBRD13:41 ;
then  F . (m + 1) turns_right m -' 1, Gauge (C,n) by A513, A900, A962;
then A1101: f turns_right m -' 1, Gauge (C,n) by A955, A932, A1098, GOBRD13:43;
A1102: a2 + 1 > p2 + 1 by A1087, NAT_1:13;
then A1103: [(p1 -' 1),p2] in Indices (Gauge (C,n)) by A948, A954, A939, A940, A1071, A1072, A1095, A1101, GOBRD13:def_6;
then A1104: 1 <= p1 -' 1 by MATRIX_1:38;
 f /. (m + 1) = (Gauge (C,n)) * ((p1 -' 1),p2) by A948, A954, A932, A939, A940, A1071, A1072, A1102, A1095, A1101, GOBRD13:def_6;
then  left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(p1 -' 1),p2) by A938, A937, A951, A939, A940, A1103, A1090, A1096, A1091, GOBRD13:23;
then  f /. (m -' 1) in left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) by A946, A947, A1072, A1076, A1100, A1087, A1104, A1091, JORDAN9:20;
then A1105: f /. (m -' 1) in (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) by A958, XBOOLE_0:def_5;
A1106: LeftComp (Rev g) is_a_component_of (L~ (Rev g)) ` by GOBOARD9:def_1;
 (left_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) c= LeftComp (Rev g) by A938, A937, A951, JORDAN9:27;
hence  C meets LeftComp (Rev g) by A653, A659, A954, A955, A958, A1105, A1099, A1106; ::_thesis: verum
end;
end;
end;
end;
end;
end;
then  C meets RightComp g by GOBOARD9:23;
hence  contradiction by A875, A884, A898, A886, JORDAN9:1, SPRECT_4:6; ::_thesis: verum
end;
then A1107: g = f /^ 0 by XREAL_1:232
.= f by FINSEQ_5:28 ;
then reconsider f = f as non constant standard special_circular_sequence ;
 F . (0 + 1) = <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))))*> by A156;
then A1108: (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) = (F . 1) /. 1 by FINSEQ_4:16
.= f /. 1 by A646, A565 ;
 F . (1 + 1) = <*((Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n)))),((Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))))*> by A156;
then A1109: (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) = (F . 2) /. 2 by FINSEQ_4:17
.= f /. 2 by A656, A565 ;
A1110: 2 < X-SpanStart (C,n) by JORDAN1H:49;
 f is clockwise_oriented
proof
 LeftComp f is_a_component_of (L~ f) ` by GOBOARD9:def_1;
then  C c= LeftComp f by A875, A886, A1107, GOBOARD9:4;
then  RightComp f misses C by GOBRD14:14, XBOOLE_1:63;
then A1111: RightComp f c= C ` by SUBSET_1:23;
 UBD (L~ f) is_outside_component_of L~ f by JORDAN2C:68;
then  UBD (L~ f) is_a_component_of (L~ f) ` by JORDAN2C:def_3;
then A1112: ( UBD (L~ f) = RightComp f or UBD (L~ f) = LeftComp f ) by JORDAN1H:24;
A1113: ((X-SpanStart (C,n)) -' 1) + 1 = X-SpanStart (C,n) by A1110, XREAL_1:235, XXREAL_0:2;
set W =  {  B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C  }  ;
A1114: Int (right_cell (f,1,(Gauge (C,n)))) c= right_cell (f,1,(Gauge (C,n))) by TOPS_1:16;
A1115: BDD C = union  {  B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C  }  by JORDAN2C:def_4;
A1116: Int (right_cell (f,1,(Gauge (C,n)))) <> {} by A652, A655, JORDAN9:9;
A1117: [((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))] in Indices (Gauge (C,n)) by A1, JORDAN11:9;
 cell ((Gauge (C,n)),((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) c= BDD C by A1, JORDAN11:6;
then  right_cell (f,1,(Gauge (C,n))) c= BDD C by A5, A439, A655, A1108, A1109, A1113, A1117, GOBRD13:26;
then A1118: Int (right_cell (f,1,(Gauge (C,n)))) c= BDD C by A1114, XBOOLE_1:1;
 Int (right_cell (f,1,(Gauge (C,n)))) c= RightComp f by A652, A655, JORDAN1H:25;
then  BDD C meets RightComp f by A1118, A1116, XBOOLE_1:68;
then consider e being set  such that
A1119: e in  {  B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C  }  and
A1120: RightComp f meets e by A1115, ZFMISC_1:80;
consider B being Subset of (TOP-REAL 2) such that
A1121: e = B and
A1122: B is_inside_component_of C by A1119;
A1123: B is bounded by A1122, JORDAN2C:def_2;
 B is_a_component_of C ` by A1122, JORDAN2C:def_2;
then  RightComp f is bounded by A1120, A1121, A1111, A1123, GOBOARD9:4, RLTOPSP1:42;
hence  f is clockwise_oriented by A1112, JORDAN1H:39, JORDAN1H:41; ::_thesis: verum
end;
then reconsider f = f as non constant standard clockwise_oriented special_circular_sequence ;
take  f ; ::_thesis: ( f is_sequence_on Gauge (C,n) & f /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) & f /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Element of NAT st 1 <= k & k + 2 <= len f holds
( ( front_right_cell (f,k,(Gauge (C,n))) misses C & front_left_cell (f,k,(Gauge (C,n))) misses C implies f turns_left k, Gauge (C,n) ) & ( front_right_cell (f,k,(Gauge (C,n))) misses C & front_left_cell (f,k,(Gauge (C,n))) meets C implies f goes_straight k, Gauge (C,n) ) & ( front_right_cell (f,k,(Gauge (C,n))) meets C implies f turns_right k, Gauge (C,n) ) ) ) )
thus  f is_sequence_on Gauge (C,n) by A439; ::_thesis: ( f /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) & f /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Element of NAT st 1 <= k & k + 2 <= len f holds
( ( front_right_cell (f,k,(Gauge (C,n))) misses C & front_left_cell (f,k,(Gauge (C,n))) misses C implies f turns_left k, Gauge (C,n) ) & ( front_right_cell (f,k,(Gauge (C,n))) misses C & front_left_cell (f,k,(Gauge (C,n))) meets C implies f goes_straight k, Gauge (C,n) ) & ( front_right_cell (f,k,(Gauge (C,n))) meets C implies f turns_right k, Gauge (C,n) ) ) ) )
thus  f /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) by A1108; ::_thesis: ( f /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Element of NAT st 1 <= k & k + 2 <= len f holds
( ( front_right_cell (f,k,(Gauge (C,n))) misses C & front_left_cell (f,k,(Gauge (C,n))) misses C implies f turns_left k, Gauge (C,n) ) & ( front_right_cell (f,k,(Gauge (C,n))) misses C & front_left_cell (f,k,(Gauge (C,n))) meets C implies f goes_straight k, Gauge (C,n) ) & ( front_right_cell (f,k,(Gauge (C,n))) meets C implies f turns_right k, Gauge (C,n) ) ) ) )
thus  f /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) by A1109; ::_thesis: for k being Element of NAT st 1 <= k & k + 2 <= len f holds
( ( front_right_cell (f,k,(Gauge (C,n))) misses C & front_left_cell (f,k,(Gauge (C,n))) misses C implies f turns_left k, Gauge (C,n) ) & ( front_right_cell (f,k,(Gauge (C,n))) misses C & front_left_cell (f,k,(Gauge (C,n))) meets C implies f goes_straight k, Gauge (C,n) ) & ( front_right_cell (f,k,(Gauge (C,n))) meets C implies f turns_right k, Gauge (C,n) ) )
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 2 <= len f implies ( ( front_right_cell (f,m,(Gauge (C,n))) misses C & front_left_cell (f,m,(Gauge (C,n))) misses C implies f turns_left m, Gauge (C,n) ) & ( front_right_cell (f,m,(Gauge (C,n))) misses C & front_left_cell (f,m,(Gauge (C,n))) meets C implies f goes_straight m, Gauge (C,n) ) & ( front_right_cell (f,m,(Gauge (C,n))) meets C implies f turns_right m, Gauge (C,n) ) ) )
assume that
A1124: 1 <= m and
A1125: m + 2 <= len f ; ::_thesis: ( ( front_right_cell (f,m,(Gauge (C,n))) misses C & front_left_cell (f,m,(Gauge (C,n))) misses C implies f turns_left m, Gauge (C,n) ) & ( front_right_cell (f,m,(Gauge (C,n))) misses C & front_left_cell (f,m,(Gauge (C,n))) meets C implies f goes_straight m, Gauge (C,n) ) & ( front_right_cell (f,m,(Gauge (C,n))) meets C implies f turns_right m, Gauge (C,n) ) )
A1126: F . ((m + 1) + 1) = f | ((m + 1) + 1) by A564, A709, A1125;
A1127: m + 1 < m + 2 by XREAL_1:6;
then A1128: f | (m + 1) = F . (m + 1) by A564, A709, A1125, XXREAL_0:2;
A1129: m + 1 <= len f by A1125, A1127, XXREAL_0:2;
then A1130: front_right_cell ((F . (m + 1)),m,(Gauge (C,n))) = front_right_cell (f,m,(Gauge (C,n))) by A652, A1124, A1128, GOBRD13:42;
A1131: m + 1 > 1 by A1124, NAT_1:13;
A1132: m = (m + 1) -' 1 by NAT_D:34;
A1133: front_left_cell ((F . (m + 1)),m,(Gauge (C,n))) = front_left_cell (f,m,(Gauge (C,n))) by A652, A1124, A1129, A1128, GOBRD13:42;
hereby  ::_thesis: ( ( front_right_cell (f,m,(Gauge (C,n))) misses C & front_left_cell (f,m,(Gauge (C,n))) meets C implies f goes_straight m, Gauge (C,n) ) & ( front_right_cell (f,m,(Gauge (C,n))) meets C implies f turns_right m, Gauge (C,n) ) )
assume that
A1134: front_right_cell (f,m,(Gauge (C,n))) misses C and
A1135: front_left_cell (f,m,(Gauge (C,n))) misses C ; ::_thesis: f turns_left m, Gauge (C,n)
 F . ((m + 1) + 1) turns_left m, Gauge (C,n) by A513, A1132, A1131, A1130, A1133, A1134, A1135;
hence  f turns_left m, Gauge (C,n) by A1124, A1125, A1126, GOBRD13:44; ::_thesis: verum
end;
hereby  ::_thesis: ( front_right_cell (f,m,(Gauge (C,n))) meets C implies f turns_right m, Gauge (C,n) )
assume that
A1136: front_right_cell (f,m,(Gauge (C,n))) misses C and
A1137: front_left_cell (f,m,(Gauge (C,n))) meets C ; ::_thesis: f goes_straight m, Gauge (C,n)
 F . ((m + 1) + 1) goes_straight m, Gauge (C,n) by A513, A1132, A1131, A1130, A1133, A1136, A1137;
hence  f goes_straight m, Gauge (C,n) by A1124, A1125, A1126, GOBRD13:45; ::_thesis: verum
end;
assume  front_right_cell (f,m,(Gauge (C,n))) meets C ; ::_thesis: f turns_right m, Gauge (C,n)
then  F . ((m + 1) + 1) turns_right m, Gauge (C,n) by A513, A1132, A1131, A1130;
hence  f turns_right m, Gauge (C,n) by A1124, A1125, A1126, GOBRD13:43; ::_thesis: verum
end;
uniqueness
 for b1, b2 being non constant standard clockwise_oriented special_circular_sequence st b1 is_sequence_on Gauge (C,n) & b1 /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) & b1 /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Element of NAT st 1 <= k & k + 2 <= len b1 holds
( ( front_right_cell (b1,k,(Gauge (C,n))) misses C & front_left_cell (b1,k,(Gauge (C,n))) misses C implies b1 turns_left k, Gauge (C,n) ) & ( front_right_cell (b1,k,(Gauge (C,n))) misses C & front_left_cell (b1,k,(Gauge (C,n))) meets C implies b1 goes_straight k, Gauge (C,n) ) & ( front_right_cell (b1,k,(Gauge (C,n))) meets C implies b1 turns_right k, Gauge (C,n) ) ) ) & b2 is_sequence_on Gauge (C,n) & b2 /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) & b2 /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Element of NAT st 1 <= k & k + 2 <= len b2 holds
( ( front_right_cell (b2,k,(Gauge (C,n))) misses C & front_left_cell (b2,k,(Gauge (C,n))) misses C implies b2 turns_left k, Gauge (C,n) ) & ( front_right_cell (b2,k,(Gauge (C,n))) misses C & front_left_cell (b2,k,(Gauge (C,n))) meets C implies b2 goes_straight k, Gauge (C,n) ) & ( front_right_cell (b2,k,(Gauge (C,n))) meets C implies b2 turns_right k, Gauge (C,n) ) ) ) holds
b1 = b2
proof
let f1, f2 be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( f1 is_sequence_on Gauge (C,n) & f1 /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) & f1 /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Element of NAT st 1 <= k & k + 2 <= len f1 holds
( ( front_right_cell (f1,k,(Gauge (C,n))) misses C & front_left_cell (f1,k,(Gauge (C,n))) misses C implies f1 turns_left k, Gauge (C,n) ) & ( front_right_cell (f1,k,(Gauge (C,n))) misses C & front_left_cell (f1,k,(Gauge (C,n))) meets C implies f1 goes_straight k, Gauge (C,n) ) & ( front_right_cell (f1,k,(Gauge (C,n))) meets C implies f1 turns_right k, Gauge (C,n) ) ) ) & f2 is_sequence_on Gauge (C,n) & f2 /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) & f2 /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Element of NAT st 1 <= k & k + 2 <= len f2 holds
( ( front_right_cell (f2,k,(Gauge (C,n))) misses C & front_left_cell (f2,k,(Gauge (C,n))) misses C implies f2 turns_left k, Gauge (C,n) ) & ( front_right_cell (f2,k,(Gauge (C,n))) misses C & front_left_cell (f2,k,(Gauge (C,n))) meets C implies f2 goes_straight k, Gauge (C,n) ) & ( front_right_cell (f2,k,(Gauge (C,n))) meets C implies f2 turns_right k, Gauge (C,n) ) ) ) implies f1 = f2 )
assume that
A1138: f1 is_sequence_on Gauge (C,n) and
A1139: f1 /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) and
A1140: f1 /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) and
A1141: for k being Element of NAT st 1 <= k & k + 2 <= len f1 holds
( ( front_right_cell (f1,k,(Gauge (C,n))) misses C & front_left_cell (f1,k,(Gauge (C,n))) misses C implies f1 turns_left k, Gauge (C,n) ) & ( front_right_cell (f1,k,(Gauge (C,n))) misses C & front_left_cell (f1,k,(Gauge (C,n))) meets C implies f1 goes_straight k, Gauge (C,n) ) & ( front_right_cell (f1,k,(Gauge (C,n))) meets C implies f1 turns_right k, Gauge (C,n) ) ) and
A1142: f2 is_sequence_on Gauge (C,n) and
A1143: f2 /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) and
A1144: f2 /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) and
A1145: for k being Element of NAT st 1 <= k & k + 2 <= len f2 holds
( ( front_right_cell (f2,k,(Gauge (C,n))) misses C & front_left_cell (f2,k,(Gauge (C,n))) misses C implies f2 turns_left k, Gauge (C,n) ) & ( front_right_cell (f2,k,(Gauge (C,n))) misses C & front_left_cell (f2,k,(Gauge (C,n))) meets C implies f2 goes_straight k, Gauge (C,n) ) & ( front_right_cell (f2,k,(Gauge (C,n))) meets C implies f2 turns_right k, Gauge (C,n) ) ) ; ::_thesis: f1 = f2
defpred S1[ Element of NAT ] means f1 | $1 = f2 | $1;
A1146: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
A1147: len f2 > 4 by GOBOARD7:34;
A1148: len f1 > 4 by GOBOARD7:34;
A1149: f1 | 1 = <*(f1 /. 1)*> by FINSEQ_5:20;
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A1150: f1 | k = f2 | k ; ::_thesis: S1[k + 1]
percases ( k = 0 or k = 1 or k > 1 ) by NAT_1:25;
suppose k = 0 ; ::_thesis: S1[k + 1]
hence  S1[k + 1] by A1139, A1143, A1149, FINSEQ_5:20; ::_thesis: verum
end;
supposeA1151: k = 1 ; ::_thesis: S1[k + 1]
 f1 | 2 = <*(f1 /. 1),(f1 /. 2)*> by A1148, FINSEQ_5:81, XXREAL_0:2;
hence  S1[k + 1] by A1139, A1140, A1143, A1144, A1147, A1151, FINSEQ_5:81, XXREAL_0:2; ::_thesis: verum
end;
supposeA1152: k > 1 ; ::_thesis: S1[k + 1]
A1153: f2 /. 1 = f2 /. (len f2) by FINSEQ_6:def_1;
A1154: f1 /. 1 = f1 /. (len f1) by FINSEQ_6:def_1;
now__::_thesis:_S1[k_+_1]
percases ( len f1 > k or k >= len f1 ) ;
supposeA1155: len f1 > k ; ::_thesis: S1[k + 1]
set m = k -' 1;
A1156: 1 <= k -' 1 by A1152, NAT_D:49;
then A1157: (k -' 1) + 1 = k by NAT_D:43;
then A1158: front_right_cell (f1,(k -' 1),(Gauge (C,n))) = front_right_cell ((f1 | k),(k -' 1),(Gauge (C,n))) by A1138, A1155, A1156, GOBRD13:42;
A1159: k + 1 <= len f1 by A1155, NAT_1:13;
A1160: now__::_thesis:_not_len_f2_<=_k
A1161: 1 < len f2 by GOBOARD7:34, XXREAL_0:2;
assume A1162: len f2 <= k ; ::_thesis: contradiction
then A1163: f2 = f2 | k by FINSEQ_1:58;
then A1164: 1 in dom (f2 | k) by FINSEQ_5:6;
 len f2 in dom (f2 | k) by A1163, FINSEQ_5:6;
then A1165: (f1 | k) /. (len f2) = f1 /. (len f2) by A1150, FINSEQ_4:70;
 len f2 <= len f1 by A1150, A1163, FINSEQ_5:16;
hence  contradiction by A1150, A1154, A1153, A1155, A1162, A1163, A1164, A1165, A1161, FINSEQ_4:70, GOBOARD7:38; ::_thesis: verum
end;
then A1166: k + 1 <= len f2 by NAT_1:13;
A1167: front_left_cell (f2,(k -' 1),(Gauge (C,n))) = front_left_cell ((f2 | k),(k -' 1),(Gauge (C,n))) by A1142, A1156, A1157, A1160, GOBRD13:42;
A1168: front_right_cell (f2,(k -' 1),(Gauge (C,n))) = front_right_cell ((f2 | k),(k -' 1),(Gauge (C,n))) by A1142, A1156, A1157, A1160, GOBRD13:42;
A1169: (k -' 1) + (1 + 1) = k + 1 by A1157;
A1170: front_left_cell (f1,(k -' 1),(Gauge (C,n))) = front_left_cell ((f1 | k),(k -' 1),(Gauge (C,n))) by A1138, A1155, A1156, A1157, GOBRD13:42;
now__::_thesis:_S1[k_+_1]
percases ( ( front_right_cell (f1,(k -' 1),(Gauge (C,n))) misses C & front_left_cell (f1,(k -' 1),(Gauge (C,n))) misses C ) or ( front_right_cell (f1,(k -' 1),(Gauge (C,n))) misses C & front_left_cell (f1,(k -' 1),(Gauge (C,n))) meets C ) or front_right_cell (f1,(k -' 1),(Gauge (C,n))) meets C ) ;
supposeA1171: ( front_right_cell (f1,(k -' 1),(Gauge (C,n))) misses C & front_left_cell (f1,(k -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: S1[k + 1]
then A1172: f1 turns_left k -' 1, Gauge (C,n) by A1141, A1156, A1159, A1169;
 f2 turns_left k -' 1, Gauge (C,n) by A1145, A1150, A1156, A1166, A1158, A1170, A1168, A1167, A1169, A1171;
hence  S1[k + 1] by A1142, A1150, A1152, A1166, A1159, A1172, GOBRD13:47; ::_thesis: verum
end;
supposeA1173: ( front_right_cell (f1,(k -' 1),(Gauge (C,n))) misses C & front_left_cell (f1,(k -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: S1[k + 1]
then A1174: f1 goes_straight k -' 1, Gauge (C,n) by A1141, A1156, A1159, A1169;
 f2 goes_straight k -' 1, Gauge (C,n) by A1145, A1150, A1156, A1166, A1158, A1170, A1168, A1167, A1169, A1173;
hence  S1[k + 1] by A1142, A1150, A1152, A1166, A1159, A1174, GOBRD13:48; ::_thesis: verum
end;
supposeA1175: front_right_cell (f1,(k -' 1),(Gauge (C,n))) meets C ; ::_thesis: S1[k + 1]
then A1176: f1 turns_right k -' 1, Gauge (C,n) by A1141, A1156, A1159, A1169;
 f2 turns_right k -' 1, Gauge (C,n) by A1145, A1150, A1156, A1166, A1158, A1168, A1169, A1175;
hence  S1[k + 1] by A1142, A1150, A1152, A1166, A1159, A1176, GOBRD13:46; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
supposeA1177: k >= len f1 ; ::_thesis: S1[k + 1]
A1178: 1 < len f1 by GOBOARD7:34, XXREAL_0:2;
A1179: f1 = f1 | k by A1177, FINSEQ_1:58;
then A1180: 1 in dom (f1 | k) by FINSEQ_5:6;
 len f1 in dom (f1 | k) by A1179, FINSEQ_5:6;
then A1181: (f2 | k) /. (len f1) = f2 /. (len f1) by A1150, FINSEQ_4:70;
 len f1 <= len f2 by A1150, A1179, FINSEQ_5:16;
then  len f2 = len f1 by A1150, A1154, A1153, A1179, A1180, A1181, A1178, FINSEQ_4:70, GOBOARD7:38;
hence  S1[k + 1] by A1150, A1177, A1179, FINSEQ_1:58; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
end;
end;
A1182: S1[ 0 ] ;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A1182, A1146);
hence  f1 = f2 by JORDAN9:2; ::_thesis: verum
end;
end;

:: deftheorem  defines Span JORDAN13:def_1_:_
 for C being non empty non horizontal non vertical being_simple_closed_curve Subset of (TOP-REAL 2)
for n being Element of NAT st n is_sufficiently_large_for C holds
for b3 being non constant standard clockwise_oriented special_circular_sequence holds
( b3 = Span (C,n) iff ( b3 is_sequence_on Gauge (C,n) & b3 /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) & b3 /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Element of NAT st 1 <= k & k + 2 <= len b3 holds
( ( front_right_cell (b3,k,(Gauge (C,n))) misses C & front_left_cell (b3,k,(Gauge (C,n))) misses C implies b3 turns_left k, Gauge (C,n) ) & ( front_right_cell (b3,k,(Gauge (C,n))) misses C & front_left_cell (b3,k,(Gauge (C,n))) meets C implies b3 goes_straight k, Gauge (C,n) ) & ( front_right_cell (b3,k,(Gauge (C,n))) meets C implies b3 turns_right k, Gauge (C,n) ) ) ) ) );