let FT be non empty RelStr ; :: thesis:
let g be FinSequence of FT; :: thesis:
let A be Subset of FT; :: thesis:
let x1, x2 be Element of FT; :: thesis:
assume A1: g is_minimum_path_in A,x1,x2 ; :: thesis:
then A2: g is continuous ;
A3: for h being FinSequence of FT st h is continuous & rng h c= A & h . 1 = x1 & h . (len h) = x2 holds
len g <= len h by A1;
A4: rng g c= A by A1;
A5: g . 1 = x1 by A1;
A6: g . (len g) = x2 by A1;
assume not g is one-to-one ; :: thesis:
then consider y1, y2 being object such that
A7: y1 in dom g and
A8: y2 in dom g and
A9: g . y1 = g . y2 and
A10: y1 <> y2 by FUNCT_1:def 4;
reconsider n1 = y1, n2 = y2 as Element of NAT by A7, A8;
A11: dom g = Seg (len g) by FINSEQ_1:def 3;
then A12: 1 <= n1 by A7, FINSEQ_1:1;
A13: n2 <= len g by A8, A11, FINSEQ_1:1;
A14: 1 <= n2 by A8, A11, FINSEQ_1:1;
A15: n1 <= len g by A7, A11, FINSEQ_1:1;

per cases by A10, XXREAL_0:1;

suppose A16: n1 > n2 ; :: thesis:

set k = (len g) -' n1;
set g2 = (g | n2) ^ (g /^ n1);
A17: len (g /^ n1) = (len g) - n1 by A15, RFINSEQ:def 1;
A18: (len g) - n1 >= 0 by A15, XREAL_1:48;
then A19: (len g) -' n1 = (len g) - n1 by XREAL_0:def 2;
A20: len (g | n2) = n2 by A13, FINSEQ_1:59;
then A21: ((g | n2) ^ (g /^ n1)) . 1 = (g | n2) . 1 by A14, FINSEQ_1:64
.= g . 1 by A14, FINSEQ_3:112 ;
A22: len ((g | n2) ^ (g /^ n1)) = (len (g | n2)) + (len (g /^ n1)) by FINSEQ_1:22
.= n2 + ((len g) - n1) by A15, A20, RFINSEQ:def 1 ;

per cases by A15, XXREAL_0:1;

suppose n1 < len g ; :: thesis:

then n1 + 1 <= len g by NAT_1:13;
then A23: (n1 + 1) - n1 <= (len g) - n1 by XREAL_1:13;
then A24: 0 + 1 <= n2 + ((len g) - n1) by XREAL_1:7;
A25: (g | n2) ^ (g /^ n1) is continuous

proof

thus len ((g | n2) ^ (g /^ n1)) >= 1 by A22, A24; :: according to FINTOPO6:def 6 :: thesis:
let i be Nat; :: thesis:
let z1 be Element of FT; :: thesis:
assume that
A26: 1 <= i and
A27: i < len ((g | n2) ^ (g /^ n1)) and
A28: z1 = ((g | n2) ^ (g /^ n1)) . i ; :: thesis:
A29: i + 1 <= len ((g | n2) ^ (g /^ n1)) by A27, NAT_1:13;
reconsider i = i as Element of NAT by ORDINAL1:def 12;
1 < i + 1 by A26, NAT_1:13;
then i + 1 in dom ((g | n2) ^ (g /^ n1)) by A29, FINSEQ_3:25;
then ((g | n2) ^ (g /^ n1)) . (i + 1) = ((g | n2) ^ (g /^ n1)) /. (i + 1) by PARTFUN1:def 6;
then reconsider z2 = ((g | n2) ^ (g /^ n1)) . (i + 1) as Element of FT ;

now :: thesis:

per cases ;

suppose A30: i < n2 ; :: thesis:

then A31: i + 1 <= n2 by NAT_1:13;
i + 1 <= len (g | n2) by A20, A30, NAT_1:13;
then A32: z2 = (g | n2) . (i + 1) by FINSEQ_1:64, NAT_1:12
.= g . (i + 1) by A31, FINSEQ_3:112 ;
A33: i < len g by A13, A30, XXREAL_0:2;
z1 = (g | n2) . i by A20, A26, A28, A30, FINSEQ_1:64
.= g . i by A30, FINSEQ_3:112 ;
hence z2 in U_FT z1 by A2, A26, A33, A32; :: thesis:

end;

suppose A34: i >= n2 ; :: thesis:

i - n2 < (n2 + ((len g) - n1)) - n2 by A22, A27, XREAL_1:9;
then A35: i -' n2 < (len g) - n1 by A34, XREAL_1:233;
then A36: (i -' n2) + n1 < ((len g) - n1) + n1 by XREAL_1:6;
A37: (len (g | n2)) + (i -' n2) = n2 + (i - n2) by A20, A34, XREAL_1:233
.= i ;

A38: now :: thesis:

per cases by A34, XXREAL_0:1;

suppose i > n2 ; :: thesis:

then A39: i - n2 > 0 by XREAL_1:50;
then i -' n2 = i - n2 by XREAL_0:def 2;
then A40: 0 + 1 <= i -' n2 by A39, NAT_1:13;
then A41: i -' n2 in dom (g /^ n1) by A17, A35, FINSEQ_3:25;
thus z1 = (g /^ n1) . (i -' n2) by A17, A28, A35, A37, A40, FINSEQ_1:65
.= g . ((i -' n2) + n1) by A15, A41, RFINSEQ:def 1 ; :: thesis:

end;

suppose A42: i = n2 ; :: thesis:

hence z1 = (g | n2) . n2 by A20, A26, A28, FINSEQ_1:64
.= g . (0 + n1) by A9, FINSEQ_3:112
.= g . ((i -' n2) + n1) by A42, XREAL_1:232 ;
:: thesis:

end;

i -' n2 < (len g) -' n1 by A15, A35, XREAL_1:233;
then (i -' n2) + 1 <= (len g) -' n1 by NAT_1:13;
then A43: (i -' n2) + 1 <= len (g /^ n1) by A15, A17, XREAL_1:233;
1 <= (i -' n2) + 1 by NAT_1:12;
then A44: (i -' n2) + 1 in dom (g /^ n1) by A43, FINSEQ_3:25;
(len (g | n2)) + ((i -' n2) + 1) = ((i - n2) + 1) + n2 by A20, A34, XREAL_1:233
.= i + 1 ;
then A45: z2 = (g /^ n1) . ((i -' n2) + 1) by A43, FINSEQ_1:65, NAT_1:12
.= g . (((i -' n2) + 1) + n1) by A15, A44, RFINSEQ:def 1
.= g . (((i -' n2) + n1) + 1) ;
1 <= (i -' n2) + n1 by A12, NAT_1:12;
hence z2 in U_FT z1 by A2, A38, A45, A36; :: thesis:

end;

hence ((g | n2) ^ (g /^ n1)) . (i + 1) in U_FT z1 ; :: thesis:

end;

A46: rng (g /^ n1) c= rng g by FINSEQ_5:33;
( rng ((g | n2) ^ (g /^ n1)) = (rng (g | n2)) \/ (rng (g /^ n1)) & rng (g | n2) c= rng g ) by FINSEQ_1:31, FINSEQ_5:19;
then A47: rng ((g | n2) ^ (g /^ n1)) c= rng g by A46, XBOOLE_1:8;
A48: ((len g) -' n1) + n1 = len g by A19;
A49: (len g) -' n1 in dom (g /^ n1) by A19, A17, A23, FINSEQ_3:25;
then ((g | n2) ^ (g /^ n1)) . ((len (g | n2)) + ((len g) -' n1)) = (g /^ n1) . ((len g) -' n1) by FINSEQ_1:def 7
.= x2 by A6, A15, A49, A48, RFINSEQ:def 1 ;
then ((g | n2) ^ (g /^ n1)) . (len ((g | n2) ^ (g /^ n1))) = x2 by A20, A22, A18, XREAL_0:def 2;
then len g <= len ((g | n2) ^ (g /^ n1)) by A4, A5, A3, A21, A47, A25, XBOOLE_1:1;
then (len g) - n2 <= (n2 + ((len g) - n1)) - n2 by A22, XREAL_1:13;
hence contradiction by A16, XREAL_1:10; :: thesis:

end;

suppose A50: n1 = len g ; :: thesis:

A51: len (g /^ n1) = (len g) - n1 by A15, RFINSEQ:def 1;
A52: (g | n2) ^ (g /^ n1) is continuous

proof

thus 1 <= len ((g | n2) ^ (g /^ n1)) by A8, A11, A22, A50, FINSEQ_1:1; :: according to FINTOPO6:def 6 :: thesis:
let i be Nat; :: thesis:
let z1 be Element of FT; :: thesis:
assume that
A53: 1 <= i and
A54: i < len ((g | n2) ^ (g /^ n1)) and
A55: z1 = ((g | n2) ^ (g /^ n1)) . i ; :: thesis:
A56: i + 1 <= len ((g | n2) ^ (g /^ n1)) by A54, NAT_1:13;
reconsider i = i as Element of NAT by ORDINAL1:def 12;
1 < i + 1 by A53, NAT_1:13;
then i + 1 in dom ((g | n2) ^ (g /^ n1)) by A56, FINSEQ_3:25;
then ((g | n2) ^ (g /^ n1)) . (i + 1) = ((g | n2) ^ (g /^ n1)) /. (i + 1) by PARTFUN1:def 6;
then reconsider z2 = ((g | n2) ^ (g /^ n1)) . (i + 1) as Element of FT ;

per cases ;

suppose A57: i < n2 ; :: thesis:

then A58: i + 1 <= n2 by NAT_1:13;
i + 1 <= len (g | n2) by A20, A57, NAT_1:13;
then A59: z2 = (g | n2) . (i + 1) by FINSEQ_1:64, NAT_1:12
.= g . (i + 1) by A58, FINSEQ_3:112 ;
A60: i < len g by A13, A57, XXREAL_0:2;
z1 = (g | n2) . i by A20, A53, A55, A57, FINSEQ_1:64
.= g . i by A57, FINSEQ_3:112 ;
hence ((g | n2) ^ (g /^ n1)) . (i + 1) in U_FT z1 by A2, A53, A60, A59; :: thesis:

end;

suppose A61: i >= n2 ; :: thesis:

i - n2 < (n2 + ((len g) - n1)) - n2 by A22, A54, XREAL_1:9;
then A62: i -' n2 < (len g) - n1 by A61, XREAL_1:233;
then A63: (i -' n2) + n1 < ((len g) - n1) + n1 by XREAL_1:6;
A64: (len (g | n2)) + (i -' n2) = n2 + (i - n2) by A20, A61, XREAL_1:233
.= i ;

A65: now :: thesis:

per cases by A61, XXREAL_0:1;

suppose i > n2 ; :: thesis:

then A66: i - n2 > 0 by XREAL_1:50;
then i -' n2 = i - n2 by XREAL_0:def 2;
then A67: 0 + 1 <= i -' n2 by A66, NAT_1:13;
then A68: i -' n2 in dom (g /^ n1) by A51, A62, FINSEQ_3:25;
thus z1 = (g /^ n1) . (i -' n2) by A51, A55, A62, A64, A67, FINSEQ_1:65
.= g . ((i -' n2) + n1) by A15, A68, RFINSEQ:def 1 ; :: thesis:

end;

suppose A69: i = n2 ; :: thesis:

hence z1 = (g | n2) . n2 by A20, A53, A55, FINSEQ_1:64
.= g . (0 + n1) by A9, FINSEQ_3:112
.= g . ((i -' n2) + n1) by A69, XREAL_1:232 ;
:: thesis:

end;

i -' n2 < (len g) -' n1 by A15, A62, XREAL_1:233;
then (i -' n2) + 1 <= (len g) -' n1 by NAT_1:13;
then A70: (i -' n2) + 1 <= len (g /^ n1) by A15, A51, XREAL_1:233;
1 <= (i -' n2) + 1 by NAT_1:12;
then A71: (i -' n2) + 1 in dom (g /^ n1) by A70, FINSEQ_3:25;
(len (g | n2)) + ((i -' n2) + 1) = ((i - n2) + 1) + n2 by A20, A61, XREAL_1:233
.= i + 1 ;
then A72: z2 = (g /^ n1) . ((i -' n2) + 1) by A70, FINSEQ_1:65, NAT_1:12
.= g . (((i -' n2) + 1) + n1) by A15, A71, RFINSEQ:def 1
.= g . (((i -' n2) + n1) + 1) ;
1 <= (i -' n2) + n1 by A12, NAT_1:12;
hence ((g | n2) ^ (g /^ n1)) . (i + 1) in U_FT z1 by A2, A65, A72, A63; :: thesis:

end;

A73: rng (g | n2) c= rng g by FINSEQ_5:19;
A74: (g | n2) ^ (g /^ n1) = (g | n2) ^ {} by A50, FINSEQ_6:167
.= g | n2 by FINSEQ_1:34 ;
then ((g | n2) ^ (g /^ n1)) . (len ((g | n2) ^ (g /^ n1))) = x2 by A6, A9, A20, A50, FINSEQ_3:112;
then len g <= len ((g | n2) ^ (g /^ n1)) by A4, A5, A3, A21, A74, A73, A52, XBOOLE_1:1;
then (len g) - n2 <= (n2 + ((len g) - n1)) - n2 by A22, XREAL_1:13;
hence contradiction by A16, XREAL_1:10; :: thesis:

end;

suppose A75: n2 > n1 ; :: thesis:

set k = (len g) -' n2;
set g2 = (g | n1) ^ (g /^ n2);
A76: len (g /^ n2) = (len g) - n2 by A13, RFINSEQ:def 1;
(len g) - n2 >= 0 by A13, XREAL_1:48;
then A77: (len g) -' n2 = (len g) - n2 by XREAL_0:def 2;
A78: len (g | n1) = n1 by A15, FINSEQ_1:59;
then A79: ((g | n1) ^ (g /^ n2)) . 1 = (g | n1) . 1 by A12, FINSEQ_1:64
.= x1 by A5, A12, FINSEQ_3:112 ;
A80: len ((g | n1) ^ (g /^ n2)) = (len (g | n1)) + (len (g /^ n2)) by FINSEQ_1:22
.= n1 + ((len g) - n2) by A13, A78, RFINSEQ:def 1 ;

per cases by A13, XXREAL_0:1;

suppose n2 < len g ; :: thesis:

then n2 + 1 <= len g by NAT_1:13;
then A81: (n2 + 1) - n2 <= (len g) - n2 by XREAL_1:13;
then A82: 0 + 1 <= n1 + ((len g) - n2) by XREAL_1:7;
A83: (g | n1) ^ (g /^ n2) is continuous

proof

thus len ((g | n1) ^ (g /^ n2)) >= 1 by A80, A82; :: according to FINTOPO6:def 6 :: thesis:
let i be Nat; :: thesis:
let z1 be Element of FT; :: thesis:
assume that
A84: 1 <= i and
A85: i < len ((g | n1) ^ (g /^ n2)) and
A86: z1 = ((g | n1) ^ (g /^ n2)) . i ; :: thesis:
A87: i + 1 <= len ((g | n1) ^ (g /^ n2)) by A85, NAT_1:13;
reconsider i = i as Element of NAT by ORDINAL1:def 12;
1 < i + 1 by A84, NAT_1:13;
then i + 1 in dom ((g | n1) ^ (g /^ n2)) by A87, FINSEQ_3:25;
then ((g | n1) ^ (g /^ n2)) . (i + 1) = ((g | n1) ^ (g /^ n2)) /. (i + 1) by PARTFUN1:def 6;
then reconsider z2 = ((g | n1) ^ (g /^ n2)) . (i + 1) as Element of FT ;

per cases ;

suppose A88: i < n1 ; :: thesis:

then A89: i + 1 <= n1 by NAT_1:13;
i + 1 <= len (g | n1) by A78, A88, NAT_1:13;
then A90: z2 = (g | n1) . (i + 1) by FINSEQ_1:64, NAT_1:12
.= g . (i + 1) by A89, FINSEQ_3:112 ;
A91: i < len g by A15, A88, XXREAL_0:2;
z1 = (g | n1) . i by A78, A84, A86, A88, FINSEQ_1:64
.= g . i by A88, FINSEQ_3:112 ;
hence ((g | n1) ^ (g /^ n2)) . (i + 1) in U_FT z1 by A2, A84, A91, A90; :: thesis:

end;

suppose A92: i >= n1 ; :: thesis:

i - n1 < (n1 + ((len g) - n2)) - n1 by A80, A85, XREAL_1:9;
then A93: i -' n1 < (len g) - n2 by A92, XREAL_1:233;
then A94: (i -' n1) + n2 < ((len g) - n2) + n2 by XREAL_1:6;
A95: (len (g | n1)) + (i -' n1) = n1 + (i - n1) by A78, A92, XREAL_1:233
.= i ;

A96: now :: thesis:

per cases by A92, XXREAL_0:1;

suppose i > n1 ; :: thesis:

then A97: i - n1 > 0 by XREAL_1:50;
then i -' n1 = i - n1 by XREAL_0:def 2;
then A98: 0 + 1 <= i -' n1 by A97, NAT_1:13;
then A99: i -' n1 in dom (g /^ n2) by A76, A93, FINSEQ_3:25;
thus z1 = (g /^ n2) . (i -' n1) by A76, A86, A93, A95, A98, FINSEQ_1:65
.= g . ((i -' n1) + n2) by A13, A99, RFINSEQ:def 1 ; :: thesis:

end;

suppose A100: i = n1 ; :: thesis:

hence z1 = (g | n1) . n1 by A78, A84, A86, FINSEQ_1:64
.= g . (0 + n2) by A9, FINSEQ_3:112
.= g . ((i -' n1) + n2) by A100, XREAL_1:232 ;
:: thesis:

end;

i -' n1 < (len g) -' n2 by A13, A93, XREAL_1:233;
then (i -' n1) + 1 <= (len g) -' n2 by NAT_1:13;
then A101: (i -' n1) + 1 <= len (g /^ n2) by A13, A76, XREAL_1:233;
1 <= (i -' n1) + 1 by NAT_1:12;
then A102: (i -' n1) + 1 in dom (g /^ n2) by A101, FINSEQ_3:25;
(len (g | n1)) + ((i -' n1) + 1) = ((i - n1) + 1) + n1 by A78, A92, XREAL_1:233
.= i + 1 ;
then A103: z2 = (g /^ n2) . ((i -' n1) + 1) by A101, FINSEQ_1:65, NAT_1:12
.= g . (((i -' n1) + 1) + n2) by A13, A102, RFINSEQ:def 1
.= g . (((i -' n1) + n2) + 1) ;
1 <= (i -' n1) + n2 by A14, NAT_1:12;
hence ((g | n1) ^ (g /^ n2)) . (i + 1) in U_FT z1 by A2, A96, A103, A94; :: thesis:

end;

A104: rng (g /^ n2) c= rng g by FINSEQ_5:33;
( rng ((g | n1) ^ (g /^ n2)) = (rng (g | n1)) \/ (rng (g /^ n2)) & rng (g | n1) c= rng g ) by FINSEQ_1:31, FINSEQ_5:19;
then rng ((g | n1) ^ (g /^ n2)) c= rng g by A104, XBOOLE_1:8;
then A105: rng ((g | n1) ^ (g /^ n2)) c= A by A4;
A106: ((len g) -' n2) + n2 = len g by A77;
A107: (len g) -' n2 in dom (g /^ n2) by A77, A76, A81, FINSEQ_3:25;
then ((g | n1) ^ (g /^ n2)) . ((len (g | n1)) + ((len g) -' n2)) = (g /^ n2) . ((len g) -' n2) by FINSEQ_1:def 7
.= x2 by A6, A13, A107, A106, RFINSEQ:def 1 ;
then ((g | n1) ^ (g /^ n2)) . (len ((g | n1) ^ (g /^ n2))) = x2 by A15, A80, A77, FINSEQ_1:59;
then len g <= len ((g | n1) ^ (g /^ n2)) by A1, A79, A105, A83;
then (len g) - n1 <= (n1 + ((len g) - n2)) - n1 by A80, XREAL_1:13;
hence contradiction by A75, XREAL_1:10; :: thesis:

end;

suppose A108: n2 = len g ; :: thesis:

A109: len (g /^ n2) = (len g) - n2 by A13, RFINSEQ:def 1;
A110: (g | n1) ^ (g /^ n2) is continuous

proof

thus len ((g | n1) ^ (g /^ n2)) >= 1 by A7, A11, A80, A108, FINSEQ_1:1; :: according to FINTOPO6:def 6 :: thesis:
let i be Nat; :: thesis:
let z1 be Element of FT; :: thesis:
assume that
A111: 1 <= i and
A112: i < len ((g | n1) ^ (g /^ n2)) and
A113: z1 = ((g | n1) ^ (g /^ n2)) . i ; :: thesis:
A114: i + 1 <= len ((g | n1) ^ (g /^ n2)) by A112, NAT_1:13;
reconsider i = i as Element of NAT by ORDINAL1:def 12;
1 < i + 1 by A111, NAT_1:13;
then i + 1 in dom ((g | n1) ^ (g /^ n2)) by A114, FINSEQ_3:25;
then ((g | n1) ^ (g /^ n2)) . (i + 1) = ((g | n1) ^ (g /^ n2)) /. (i + 1) by PARTFUN1:def 6;
then reconsider z2 = ((g | n1) ^ (g /^ n2)) . (i + 1) as Element of FT ;

per cases ;

suppose A115: i < n1 ; :: thesis:

then A116: i + 1 <= n1 by NAT_1:13;
i + 1 <= len (g | n1) by A78, A115, NAT_1:13;
then A117: z2 = (g | n1) . (i + 1) by FINSEQ_1:64, NAT_1:12
.= g . (i + 1) by A116, FINSEQ_3:112 ;
A118: i < len g by A15, A115, XXREAL_0:2;
z1 = (g | n1) . i by A78, A111, A113, A115, FINSEQ_1:64
.= g . i by A115, FINSEQ_3:112 ;
hence ((g | n1) ^ (g /^ n2)) . (i + 1) in U_FT z1 by A2, A111, A118, A117; :: thesis:

end;

suppose A119: i >= n1 ; :: thesis:

i - n1 < (n1 + ((len g) - n2)) - n1 by A80, A112, XREAL_1:9;
then A120: i -' n1 < (len g) - n2 by A119, XREAL_1:233;
then A121: (i -' n1) + n2 < ((len g) - n2) + n2 by XREAL_1:6;
A122: (len (g | n1)) + (i -' n1) = n1 + (i - n1) by A78, A119, XREAL_1:233
.= i ;

A123: now :: thesis:

per cases by A119, XXREAL_0:1;

suppose i > n1 ; :: thesis:

then A124: i - n1 > 0 by XREAL_1:50;
then i -' n1 = i - n1 by XREAL_0:def 2;
then A125: 0 + 1 <= i -' n1 by A124, NAT_1:13;
then A126: i -' n1 in dom (g /^ n2) by A109, A120, FINSEQ_3:25;
thus z1 = (g /^ n2) . (i -' n1) by A109, A113, A120, A122, A125, FINSEQ_1:65
.= g . ((i -' n1) + n2) by A13, A126, RFINSEQ:def 1 ; :: thesis:

end;

suppose A127: i = n1 ; :: thesis:

hence z1 = (g | n1) . n1 by A78, A111, A113, FINSEQ_1:64
.= g . (0 + n2) by A9, FINSEQ_3:112
.= g . ((i -' n1) + n2) by A127, XREAL_1:232 ;
:: thesis:

end;

i -' n1 < (len g) -' n2 by A13, A120, XREAL_1:233;
then (i -' n1) + 1 <= (len g) -' n2 by NAT_1:13;
then A128: (i -' n1) + 1 <= len (g /^ n2) by A13, A109, XREAL_1:233;
1 <= (i -' n1) + 1 by NAT_1:12;
then A129: (i -' n1) + 1 in dom (g /^ n2) by A128, FINSEQ_3:25;
(len (g | n1)) + ((i -' n1) + 1) = ((i - n1) + 1) + n1 by A78, A119, XREAL_1:233
.= i + 1 ;
then A130: z2 = (g /^ n2) . ((i -' n1) + 1) by A128, FINSEQ_1:65, NAT_1:12
.= g . (((i -' n1) + 1) + n2) by A13, A129, RFINSEQ:def 1
.= g . (((i -' n1) + n2) + 1) ;
1 <= (i -' n1) + n2 by A14, NAT_1:12;
hence ((g | n1) ^ (g /^ n2)) . (i + 1) in U_FT z1 by A2, A123, A130, A121; :: thesis:

end;

A131: (g | n1) ^ (g /^ n2) = (g | n1) ^ {} by A108, FINSEQ_6:167
.= g | n1 by FINSEQ_1:34 ;
rng (g | n1) c= rng g by FINSEQ_5:19;
then A132: rng ((g | n1) ^ (g /^ n2)) c= A by A4, A131;
((g | n1) ^ (g /^ n2)) . (len ((g | n1) ^ (g /^ n2))) = x2 by A6, A9, A78, A108, A131, FINSEQ_3:112;
then len g <= len ((g | n1) ^ (g /^ n2)) by A1, A79, A132, A110;
then (len g) - n1 <= (n1 + ((len g) - n2)) - n1 by A80, XREAL_1:13;
hence contradiction by A75, XREAL_1:10; :: thesis:

end;