let f be V22() standard clockwise_oriented special_circular_sequence; :: thesis: for G being Go-board

for k being Nat st f is_sequence_on G & 1 <= k & k + 1 <= len f & f /. k = E-max (L~ f) holds

ex i, j being Nat st

( [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) )

let G be Go-board; :: thesis: for k being Nat st f is_sequence_on G & 1 <= k & k + 1 <= len f & f /. k = E-max (L~ f) holds

ex i, j being Nat st

( [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) )

let k be Nat; :: thesis: ( f is_sequence_on G & 1 <= k & k + 1 <= len f & f /. k = E-max (L~ f) implies ex i, j being Nat st

( [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) ) )

assume that

A1: f is_sequence_on G and

A2: 1 <= k and

A3: k + 1 <= len f and

A4: f /. k = E-max (L~ f) ; :: thesis: ex i, j being Nat st

( [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) )

consider i1, j1, i2, j2 being Nat such that

A5: [i1,j1] in Indices G and

A6: f /. k = G * (i1,j1) and

A7: [i2,j2] in Indices G and

A8: f /. (k + 1) = G * (i2,j2) and

A9: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A2, A3, JORDAN8:3;

A10: (G * (i1,j1)) `1 = E-bound (L~ f) by A4, A6, EUCLID:52;

A11: j2 <= width G by A7, MATRIX_0:32;

take i2 ; :: thesis: ex j being Nat st

( [i2,(j + 1)] in Indices G & [i2,j] in Indices G & f /. k = G * (i2,(j + 1)) & f /. (k + 1) = G * (i2,j) )

take j2 ; :: thesis: ( [i2,(j2 + 1)] in Indices G & [i2,j2] in Indices G & f /. k = G * (i2,(j2 + 1)) & f /. (k + 1) = G * (i2,j2) )

A12: i1 <= len G by A5, MATRIX_0:32;

A13: k + 1 >= 1 + 1 by A2, XREAL_1:7;

then A14: len f >= 2 by A3, XXREAL_0:2;

k + 1 >= 1 by NAT_1:11;

then A15: k + 1 in dom f by A3, FINSEQ_3:25;

then f /. (k + 1) in L~ f by A3, A13, GOBOARD1:1, XXREAL_0:2;

then A16: (G * (i1,j1)) `1 >= (G * (i2,j2)) `1 by A8, A10, PSCOMP_1:24;

A17: ( 1 <= i1 & 1 <= j1 ) by A5, MATRIX_0:32;

A18: k < len f by A3, NAT_1:13;

then A19: k in dom f by A2, FINSEQ_3:25;

A20: i2 <= len G by A7, MATRIX_0:32;

A21: j1 <= width G by A5, MATRIX_0:32;

A22: ( 1 <= i2 & 1 <= j2 ) by A7, MATRIX_0:32;

thus f /. (k + 1) = G * (i2,j2) by A8; :: thesis: verum

for k being Nat st f is_sequence_on G & 1 <= k & k + 1 <= len f & f /. k = E-max (L~ f) holds

ex i, j being Nat st

( [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) )

let G be Go-board; :: thesis: for k being Nat st f is_sequence_on G & 1 <= k & k + 1 <= len f & f /. k = E-max (L~ f) holds

ex i, j being Nat st

( [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) )

let k be Nat; :: thesis: ( f is_sequence_on G & 1 <= k & k + 1 <= len f & f /. k = E-max (L~ f) implies ex i, j being Nat st

( [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) ) )

assume that

A1: f is_sequence_on G and

A2: 1 <= k and

A3: k + 1 <= len f and

A4: f /. k = E-max (L~ f) ; :: thesis: ex i, j being Nat st

( [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) )

consider i1, j1, i2, j2 being Nat such that

A5: [i1,j1] in Indices G and

A6: f /. k = G * (i1,j1) and

A7: [i2,j2] in Indices G and

A8: f /. (k + 1) = G * (i2,j2) and

A9: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A2, A3, JORDAN8:3;

A10: (G * (i1,j1)) `1 = E-bound (L~ f) by A4, A6, EUCLID:52;

A11: j2 <= width G by A7, MATRIX_0:32;

take i2 ; :: thesis: ex j being Nat st

( [i2,(j + 1)] in Indices G & [i2,j] in Indices G & f /. k = G * (i2,(j + 1)) & f /. (k + 1) = G * (i2,j) )

take j2 ; :: thesis: ( [i2,(j2 + 1)] in Indices G & [i2,j2] in Indices G & f /. k = G * (i2,(j2 + 1)) & f /. (k + 1) = G * (i2,j2) )

A12: i1 <= len G by A5, MATRIX_0:32;

A13: k + 1 >= 1 + 1 by A2, XREAL_1:7;

then A14: len f >= 2 by A3, XXREAL_0:2;

k + 1 >= 1 by NAT_1:11;

then A15: k + 1 in dom f by A3, FINSEQ_3:25;

then f /. (k + 1) in L~ f by A3, A13, GOBOARD1:1, XXREAL_0:2;

then A16: (G * (i1,j1)) `1 >= (G * (i2,j2)) `1 by A8, A10, PSCOMP_1:24;

A17: ( 1 <= i1 & 1 <= j1 ) by A5, MATRIX_0:32;

A18: k < len f by A3, NAT_1:13;

then A19: k in dom f by A2, FINSEQ_3:25;

A20: i2 <= len G by A7, MATRIX_0:32;

A21: j1 <= width G by A5, MATRIX_0:32;

A22: ( 1 <= i2 & 1 <= j2 ) by A7, MATRIX_0:32;

now :: thesis: ( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) )end;

hence
( [i2,(j2 + 1)] in Indices G & [i2,j2] in Indices G & f /. k = G * (i2,(j2 + 1)) )
; :: thesis: f /. (k + 1) = G * (i2,j2)per cases
( ( i1 = i2 & j2 + 1 = j1 ) or ( i2 + 1 = i1 & j2 = j1 & k <> 1 ) or ( i2 + 1 = i1 & j2 = j1 & k = 1 ) or ( i2 = i1 + 1 & j1 = j2 ) or ( i1 = i2 & j2 = j1 + 1 ) )
by A9;

end;

suppose
( i1 = i2 & j2 + 1 = j1 )
; :: thesis: ( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) )

hence
( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) )
by A5, A6, A7; :: thesis: verum

end;suppose A23:
( i2 + 1 = i1 & j2 = j1 & k <> 1 )
; :: thesis: ( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) )

reconsider k9 = k - 1 as Nat by A19, FINSEQ_3:26;

k > 1 by A2, A23, XXREAL_0:1;

then k >= 1 + 1 by NAT_1:13;

then A24: k9 >= (1 + 1) - 1 by XREAL_1:9;

then consider i3, j3, i4, j4 being Nat such that

A25: [i3,j3] in Indices G and

A26: f /. k9 = G * (i3,j3) and

A27: [i4,j4] in Indices G and

A28: f /. (k9 + 1) = G * (i4,j4) and

A29: ( ( i3 = i4 & j3 + 1 = j4 ) or ( i3 + 1 = i4 & j3 = j4 ) or ( i3 = i4 + 1 & j3 = j4 ) or ( i3 = i4 & j3 = j4 + 1 ) ) by A1, A18, JORDAN8:3;

A30: i1 = i4 by A5, A6, A27, A28, GOBOARD1:5;

k9 + 1 < len f by A3, NAT_1:13;

then k9 < len f by NAT_1:13;

then A31: k9 in dom f by A24, FINSEQ_3:25;

A32: i3 <= len G by A25, MATRIX_0:32;

A33: j1 = j4 by A5, A6, A27, A28, GOBOARD1:5;

A34: j3 <= width G by A25, MATRIX_0:32;

A35: ( 1 <= i3 & 1 <= j3 ) by A25, MATRIX_0:32;

A36: j3 = j4

f /. k9 in L~ f by A3, A13, A31, GOBOARD1:1, XXREAL_0:2;

then A41: (G * (i1,j1)) `1 >= (G * (i3,j3)) `1 by A10, A26, PSCOMP_1:24;

end;k > 1 by A2, A23, XXREAL_0:1;

then k >= 1 + 1 by NAT_1:13;

then A24: k9 >= (1 + 1) - 1 by XREAL_1:9;

then consider i3, j3, i4, j4 being Nat such that

A25: [i3,j3] in Indices G and

A26: f /. k9 = G * (i3,j3) and

A27: [i4,j4] in Indices G and

A28: f /. (k9 + 1) = G * (i4,j4) and

A29: ( ( i3 = i4 & j3 + 1 = j4 ) or ( i3 + 1 = i4 & j3 = j4 ) or ( i3 = i4 + 1 & j3 = j4 ) or ( i3 = i4 & j3 = j4 + 1 ) ) by A1, A18, JORDAN8:3;

A30: i1 = i4 by A5, A6, A27, A28, GOBOARD1:5;

k9 + 1 < len f by A3, NAT_1:13;

then k9 < len f by NAT_1:13;

then A31: k9 in dom f by A24, FINSEQ_3:25;

A32: i3 <= len G by A25, MATRIX_0:32;

A33: j1 = j4 by A5, A6, A27, A28, GOBOARD1:5;

A34: j3 <= width G by A25, MATRIX_0:32;

A35: ( 1 <= i3 & 1 <= j3 ) by A25, MATRIX_0:32;

A36: j3 = j4

proof

A40:
k9 + 1 = k
;
assume A37:
j3 <> j4
; :: thesis: contradiction

end;per cases
( ( i3 = i4 & j4 = j3 + 1 ) or ( i3 = i4 & j4 + 1 = j3 ) )
by A29, A37;

end;

suppose A38:
( i3 = i4 & j4 = j3 + 1 )
; :: thesis: contradiction

then
(G * (i3,j3)) `1 <> E-bound (L~ f)
by A1, A18, A24, A25, A26, A27, A28, Th18;

then (G * (i3,1)) `1 <> E-bound (L~ f) by A32, A35, A34, GOBOARD5:2;

then (E-max (L~ f)) `1 <> E-bound (L~ f) by A4, A6, A12, A17, A21, A30, A38, GOBOARD5:2;

hence contradiction by EUCLID:52; :: thesis: verum

end;then (G * (i3,1)) `1 <> E-bound (L~ f) by A32, A35, A34, GOBOARD5:2;

then (E-max (L~ f)) `1 <> E-bound (L~ f) by A4, A6, A12, A17, A21, A30, A38, GOBOARD5:2;

hence contradiction by EUCLID:52; :: thesis: verum

suppose A39:
( i3 = i4 & j4 + 1 = j3 )
; :: thesis: contradiction

(G * (i3,j3)) `1 =
(G * (i3,1)) `1
by A32, A35, A34, GOBOARD5:2

.= (E-max (L~ f)) `1 by A4, A6, A12, A17, A21, A30, A39, GOBOARD5:2

.= E-bound (L~ f) by EUCLID:52 ;

then G * (i3,j3) in E-most (L~ f) by A14, A26, A31, GOBOARD1:1, SPRECT_2:13;

then (G * (i4,j4)) `2 >= (G * (i3,j3)) `2 by A4, A28, PSCOMP_1:47;

then j4 >= j4 + 1 by A12, A17, A30, A33, A34, A39, GOBOARD5:4;

hence contradiction by NAT_1:13; :: thesis: verum

end;.= (E-max (L~ f)) `1 by A4, A6, A12, A17, A21, A30, A39, GOBOARD5:2

.= E-bound (L~ f) by EUCLID:52 ;

then G * (i3,j3) in E-most (L~ f) by A14, A26, A31, GOBOARD1:1, SPRECT_2:13;

then (G * (i4,j4)) `2 >= (G * (i3,j3)) `2 by A4, A28, PSCOMP_1:47;

then j4 >= j4 + 1 by A12, A17, A30, A33, A34, A39, GOBOARD5:4;

hence contradiction by NAT_1:13; :: thesis: verum

f /. k9 in L~ f by A3, A13, A31, GOBOARD1:1, XXREAL_0:2;

then A41: (G * (i1,j1)) `1 >= (G * (i3,j3)) `1 by A10, A26, PSCOMP_1:24;

now :: thesis: contradictionend;

hence
( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) )
; :: thesis: verumper cases
( i4 + 1 = i3 or i4 = i3 + 1 )
by A29, A36;

end;

suppose
i4 + 1 = i3
; :: thesis: contradiction

then
i4 >= i4 + 1
by A17, A21, A30, A33, A32, A41, A36, GOBOARD5:3;

hence contradiction by NAT_1:13; :: thesis: verum

end;hence contradiction by NAT_1:13; :: thesis: verum

suppose A42:
i4 = i3 + 1
; :: thesis: contradiction

k9 + (1 + 1) <= len f
by A3;

then A43: (LSeg (f,k9)) /\ (LSeg (f,k)) = {(f /. k)} by A24, A40, TOPREAL1:def 6;

( f /. k9 in LSeg (f,k9) & f /. (k + 1) in LSeg (f,k) ) by A2, A3, A18, A24, A40, TOPREAL1:21;

then f /. (k + 1) in {(f /. k)} by A8, A23, A26, A30, A33, A36, A42, A43, XBOOLE_0:def 4;

then A44: f /. (k + 1) = f /. k by TARSKI:def 1;

i1 <> i2 by A23;

hence contradiction by A5, A6, A7, A8, A44, GOBOARD1:5; :: thesis: verum

end;then A43: (LSeg (f,k9)) /\ (LSeg (f,k)) = {(f /. k)} by A24, A40, TOPREAL1:def 6;

( f /. k9 in LSeg (f,k9) & f /. (k + 1) in LSeg (f,k) ) by A2, A3, A18, A24, A40, TOPREAL1:21;

then f /. (k + 1) in {(f /. k)} by A8, A23, A26, A30, A33, A36, A42, A43, XBOOLE_0:def 4;

then A44: f /. (k + 1) = f /. k by TARSKI:def 1;

i1 <> i2 by A23;

hence contradiction by A5, A6, A7, A8, A44, GOBOARD1:5; :: thesis: verum

suppose A45:
( i2 + 1 = i1 & j2 = j1 & k = 1 )
; :: thesis: ( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) )

set k1 = len f;

k < len f by A3, NAT_1:13;

then A46: len f > 1 + 0 by A2, XXREAL_0:2;

then len f in dom f by FINSEQ_3:25;

then reconsider k9 = (len f) - 1 as Nat by FINSEQ_3:26;

k + 1 >= 1 + 1 by A2, XREAL_1:7;

then len f >= 1 + 1 by A3, XXREAL_0:2;

then A47: k9 >= (1 + 1) - 1 by XREAL_1:9;

then consider i3, j3, i4, j4 being Nat such that

A48: [i3,j3] in Indices G and

A49: f /. k9 = G * (i3,j3) and

A50: [i4,j4] in Indices G and

A51: f /. (k9 + 1) = G * (i4,j4) and

A52: ( ( i3 = i4 & j3 + 1 = j4 ) or ( i3 + 1 = i4 & j3 = j4 ) or ( i3 = i4 + 1 & j3 = j4 ) or ( i3 = i4 & j3 = j4 + 1 ) ) by A1, JORDAN8:3;

A53: f /. (len f) = f /. 1 by FINSEQ_6:def 1;

then A54: i1 = i4 by A5, A6, A45, A50, A51, GOBOARD1:5;

A55: j1 = j4 by A5, A6, A45, A53, A50, A51, GOBOARD1:5;

A56: j3 <= width G by A48, MATRIX_0:32;

k9 + 1 <= len f ;

then k9 < len f by NAT_1:13;

then A57: k9 in dom f by A47, FINSEQ_3:25;

then f /. k9 in L~ f by A3, A13, GOBOARD1:1, XXREAL_0:2;

then A58: (G * (i1,j1)) `1 >= (G * (i3,j3)) `1 by A10, A49, PSCOMP_1:24;

A59: i3 <= len G by A48, MATRIX_0:32;

A60: ( 1 <= i3 & 1 <= j3 ) by A48, MATRIX_0:32;

A61: j3 = j4

end;k < len f by A3, NAT_1:13;

then A46: len f > 1 + 0 by A2, XXREAL_0:2;

then len f in dom f by FINSEQ_3:25;

then reconsider k9 = (len f) - 1 as Nat by FINSEQ_3:26;

k + 1 >= 1 + 1 by A2, XREAL_1:7;

then len f >= 1 + 1 by A3, XXREAL_0:2;

then A47: k9 >= (1 + 1) - 1 by XREAL_1:9;

then consider i3, j3, i4, j4 being Nat such that

A48: [i3,j3] in Indices G and

A49: f /. k9 = G * (i3,j3) and

A50: [i4,j4] in Indices G and

A51: f /. (k9 + 1) = G * (i4,j4) and

A52: ( ( i3 = i4 & j3 + 1 = j4 ) or ( i3 + 1 = i4 & j3 = j4 ) or ( i3 = i4 + 1 & j3 = j4 ) or ( i3 = i4 & j3 = j4 + 1 ) ) by A1, JORDAN8:3;

A53: f /. (len f) = f /. 1 by FINSEQ_6:def 1;

then A54: i1 = i4 by A5, A6, A45, A50, A51, GOBOARD1:5;

A55: j1 = j4 by A5, A6, A45, A53, A50, A51, GOBOARD1:5;

A56: j3 <= width G by A48, MATRIX_0:32;

k9 + 1 <= len f ;

then k9 < len f by NAT_1:13;

then A57: k9 in dom f by A47, FINSEQ_3:25;

then f /. k9 in L~ f by A3, A13, GOBOARD1:1, XXREAL_0:2;

then A58: (G * (i1,j1)) `1 >= (G * (i3,j3)) `1 by A10, A49, PSCOMP_1:24;

A59: i3 <= len G by A48, MATRIX_0:32;

A60: ( 1 <= i3 & 1 <= j3 ) by A48, MATRIX_0:32;

A61: j3 = j4

proof

A65:
k9 + 1 = len f
;
assume A62:
j3 <> j4
; :: thesis: contradiction

end;per cases
( ( i3 = i4 & j4 = j3 + 1 ) or ( i3 = i4 & j4 + 1 = j3 ) )
by A52, A62;

end;

suppose A63:
( i3 = i4 & j4 = j3 + 1 )
; :: thesis: contradiction

then
(G * (i3,j3)) `1 <> E-bound (L~ f)
by A1, A47, A48, A49, A50, A51, Th18;

then (G * (i3,1)) `1 <> E-bound (L~ f) by A59, A60, A56, GOBOARD5:2;

then (E-max (L~ f)) `1 <> E-bound (L~ f) by A4, A6, A12, A17, A21, A54, A63, GOBOARD5:2;

hence contradiction by EUCLID:52; :: thesis: verum

end;then (G * (i3,1)) `1 <> E-bound (L~ f) by A59, A60, A56, GOBOARD5:2;

then (E-max (L~ f)) `1 <> E-bound (L~ f) by A4, A6, A12, A17, A21, A54, A63, GOBOARD5:2;

hence contradiction by EUCLID:52; :: thesis: verum

suppose A64:
( i3 = i4 & j4 + 1 = j3 )
; :: thesis: contradiction

(G * (i3,j3)) `1 =
(G * (i3,1)) `1
by A59, A60, A56, GOBOARD5:2

.= (E-max (L~ f)) `1 by A4, A6, A12, A17, A21, A54, A64, GOBOARD5:2

.= E-bound (L~ f) by EUCLID:52 ;

then G * (i3,j3) in E-most (L~ f) by A14, A49, A57, GOBOARD1:1, SPRECT_2:13;

then (G * (i4,j4)) `2 >= (G * (i3,j3)) `2 by A4, A45, A53, A51, PSCOMP_1:47;

then j4 >= j4 + 1 by A12, A17, A54, A55, A56, A64, GOBOARD5:4;

hence contradiction by NAT_1:13; :: thesis: verum

end;.= (E-max (L~ f)) `1 by A4, A6, A12, A17, A21, A54, A64, GOBOARD5:2

.= E-bound (L~ f) by EUCLID:52 ;

then G * (i3,j3) in E-most (L~ f) by A14, A49, A57, GOBOARD1:1, SPRECT_2:13;

then (G * (i4,j4)) `2 >= (G * (i3,j3)) `2 by A4, A45, A53, A51, PSCOMP_1:47;

then j4 >= j4 + 1 by A12, A17, A54, A55, A56, A64, GOBOARD5:4;

hence contradiction by NAT_1:13; :: thesis: verum

now :: thesis: contradictionend;

hence
( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) )
; :: thesis: verumper cases
( i4 + 1 = i3 or i4 = i3 + 1 )
by A52, A61;

end;

suppose
i4 + 1 = i3
; :: thesis: contradiction

then
i4 >= i4 + 1
by A17, A21, A54, A55, A59, A58, A61, GOBOARD5:3;

hence contradiction by NAT_1:13; :: thesis: verum

end;hence contradiction by NAT_1:13; :: thesis: verum

suppose A66:
i4 = i3 + 1
; :: thesis: contradiction

(len f) - 1 >= 0
by A46, XREAL_1:19;

then (len f) -' 1 = (len f) - 1 by XREAL_0:def 2;

then A67: (LSeg (f,k)) /\ (LSeg (f,k9)) = {(f . k)} by A45, JORDAN4:42

.= {(f /. k)} by A19, PARTFUN1:def 6 ;

( f /. k9 in LSeg (f,k9) & f /. (k + 1) in LSeg (f,k) ) by A2, A3, A47, A65, TOPREAL1:21;

then f /. (k + 1) in {(f /. k)} by A8, A45, A49, A54, A55, A61, A66, A67, XBOOLE_0:def 4;

then A68: f /. (k + 1) = f /. k by TARSKI:def 1;

i1 <> i2 by A45;

hence contradiction by A5, A6, A7, A8, A68, GOBOARD1:5; :: thesis: verum

end;then (len f) -' 1 = (len f) - 1 by XREAL_0:def 2;

then A67: (LSeg (f,k)) /\ (LSeg (f,k9)) = {(f . k)} by A45, JORDAN4:42

.= {(f /. k)} by A19, PARTFUN1:def 6 ;

( f /. k9 in LSeg (f,k9) & f /. (k + 1) in LSeg (f,k) ) by A2, A3, A47, A65, TOPREAL1:21;

then f /. (k + 1) in {(f /. k)} by A8, A45, A49, A54, A55, A61, A66, A67, XBOOLE_0:def 4;

then A68: f /. (k + 1) = f /. k by TARSKI:def 1;

i1 <> i2 by A45;

hence contradiction by A5, A6, A7, A8, A68, GOBOARD1:5; :: thesis: verum

suppose
( i2 = i1 + 1 & j1 = j2 )
; :: thesis: ( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) )

then
i1 >= i1 + 1
by A17, A21, A20, A16, GOBOARD5:3;

hence ( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) ) by NAT_1:13; :: thesis: verum

end;hence ( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) ) by NAT_1:13; :: thesis: verum

suppose A69:
( i1 = i2 & j2 = j1 + 1 )
; :: thesis: ( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) )

(G * (i2,j2)) `1 =
(G * (i2,1)) `1
by A20, A22, A11, GOBOARD5:2

.= E-bound (L~ f) by A12, A17, A21, A10, A69, GOBOARD5:2 ;

then G * (i2,j2) in E-most (L~ f) by A8, A14, A15, GOBOARD1:1, SPRECT_2:13;

then (G * (i1,j1)) `2 >= (G * (i2,j2)) `2 by A4, A6, PSCOMP_1:47;

then j1 >= j1 + 1 by A12, A17, A11, A69, GOBOARD5:4;

hence ( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) ) by NAT_1:13; :: thesis: verum

end;.= E-bound (L~ f) by A12, A17, A21, A10, A69, GOBOARD5:2 ;

then G * (i2,j2) in E-most (L~ f) by A8, A14, A15, GOBOARD1:1, SPRECT_2:13;

then (G * (i1,j1)) `2 >= (G * (i2,j2)) `2 by A4, A6, PSCOMP_1:47;

then j1 >= j1 + 1 by A12, A17, A11, A69, GOBOARD5:4;

hence ( [i2,j2] in Indices G & [i2,(j2 + 1)] in Indices G & f /. k = G * (i2,(j2 + 1)) ) by NAT_1:13; :: thesis: verum

thus f /. (k + 1) = G * (i2,j2) by A8; :: thesis: verum