let k be Element of NAT ; :: thesis: for D being set
for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_left k -' 1,G & f2 turns_left k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let D be set ; :: thesis: for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_left k -' 1,G & f2 turns_left k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let f1, f2 be FinSequence of D; :: thesis: for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_left k -' 1,G & f2 turns_left k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let G be Matrix of D; :: thesis: ( 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_left k -' 1,G & f2 turns_left k -' 1,G implies f1 | (k + 1) = f2 | (k + 1) )
assume that
A1: ( 1 < k & k + 1 <= len f1 ) and
A2: k + 1 <= len f2 and
A3: f1 is_sequence_on G and
A4: f1 | k = f2 | k and
A5: f1 turns_left k -' 1,G and
A6: f2 turns_left k -' 1,G ; :: thesis: f1 | (k + 1) = f2 | (k + 1)
A7: 1 <= k -' 1 by A1, NAT_D:49;
A8: k = (k -' 1) + 1 by A1, XREAL_1:237;
A9: k <= k + 1 by NAT_1:12;
then A10: ( k <= len f1 & k <= len f2 ) by A1, A2, XXREAL_0:2;
A11: k <= len (f1 | k) by A1, A9, FINSEQ_1:80, XXREAL_0:2;
k -' 1 <= k by NAT_D:35;
then k -' 1 <= len (f1 | k) by A1, A9, FINSEQ_1:80, XXREAL_0:2;
then A12: ( k -' 1 in dom (f1 | k) & k in dom (f1 | k) ) by A1, A7, A11, FINSEQ_3:27;
then A13: ( f1 /. (k -' 1) = (f1 | k) /. (k -' 1) & f1 /. k = (f1 | k) /. k ) by FINSEQ_4:85;
A14: ( f2 /. (k -' 1) = (f2 | k) /. (k -' 1) & f2 /. k = (f2 | k) /. k ) by A4, A12, FINSEQ_4:85;
A15: k + 1 = (k -' 1) + (1 + 1) by A8;
consider i1, j1, i2, j2 being Element of NAT such that
A16: ( [i1,j1] in Indices G & f1 /. (k -' 1) = G * i1,j1 ) and
A17: ( [i2,j2] in Indices G & f1 /. k = G * i2,j2 ) and
A18: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A3, A7, A8, A10, JORDAN8:6;
A19: ( i1 + 1 > i1 & i2 + 1 > i2 & j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;

now

per cases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A18;

suppose A20: ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis: f1 /. (k + 1) = f2 /. (k + 1)

hence f1 /. (k + 1) = G * (i2 -' 1),j2 by A5, A8, A15, A16, A17, A19, Def7
.= f2 /. (k + 1) by A4, A6, A8, A13, A14, A15, A16, A17, A19, A20, Def7 ;
:: thesis: verum

end;

suppose A21: ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis: f1 /. (k + 1) = f2 /. (k + 1)

hence f1 /. (k + 1) = G * i2,(j2 + 1) by A5, A8, A15, A16, A17, A19, Def7
.= f2 /. (k + 1) by A4, A6, A8, A13, A14, A15, A16, A17, A19, A21, Def7 ;
:: thesis: verum

end;

suppose A22: ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis: f1 /. (k + 1) = f2 /. (k + 1)

hence f1 /. (k + 1) = G * i2,(j2 -' 1) by A5, A8, A15, A16, A17, A19, Def7
.= f2 /. (k + 1) by A4, A6, A8, A13, A14, A15, A16, A17, A19, A22, Def7 ;
:: thesis: verum

end;

suppose A23: ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis: f1 /. (k + 1) = f2 /. (k + 1)

hence f1 /. (k + 1) = G * (i2 + 1),j2 by A5, A8, A15, A16, A17, A19, Def7
.= f2 /. (k + 1) by A4, A6, A8, A13, A14, A15, A16, A17, A19, A23, Def7 ;
:: thesis: verum

end;

end;

end;

hence f1 | (k + 1) = (f2 | k) ^ <*(f2 /. (k + 1))*> by A1, A4, JORDAN8:2
.= f2 | (k + 1) by A2, JORDAN8:2 ;
:: thesis: verum