let h be non constant standard special_circular_sequence; :: thesis: for I, i1, i being Nat

for Y being non empty finite Subset of NAT st 1 <= i & i <= len h & 1 <= I & I <= width (GoB h) & Y = { j where j is Element of NAT : ( [j,I] in Indices (GoB h) & ex k being Nat st

( k in dom h & h /. k = (GoB h) * (j,I) ) ) } & (h /. i) `2 = ((GoB h) * (1,I)) `2 & i1 = min Y holds

((GoB h) * (i1,I)) `1 <= (h /. i) `1

let I, i1, i be Nat; :: thesis: for Y being non empty finite Subset of NAT st 1 <= i & i <= len h & 1 <= I & I <= width (GoB h) & Y = { j where j is Element of NAT : ( [j,I] in Indices (GoB h) & ex k being Nat st

( k in dom h & h /. k = (GoB h) * (j,I) ) ) } & (h /. i) `2 = ((GoB h) * (1,I)) `2 & i1 = min Y holds

((GoB h) * (i1,I)) `1 <= (h /. i) `1

let Y be non empty finite Subset of NAT; :: thesis: ( 1 <= i & i <= len h & 1 <= I & I <= width (GoB h) & Y = { j where j is Element of NAT : ( [j,I] in Indices (GoB h) & ex k being Nat st

( k in dom h & h /. k = (GoB h) * (j,I) ) ) } & (h /. i) `2 = ((GoB h) * (1,I)) `2 & i1 = min Y implies ((GoB h) * (i1,I)) `1 <= (h /. i) `1 )

A1: h /. i = |[((h /. i) `1),((h /. i) `2)]| by EUCLID:53;

assume A2: ( 1 <= i & i <= len h & 1 <= I & I <= width (GoB h) & Y = { j where j is Element of NAT : ( [j,I] in Indices (GoB h) & ex k being Nat st

( k in dom h & h /. k = (GoB h) * (j,I) ) ) } & (h /. i) `2 = ((GoB h) * (1,I)) `2 & i1 = min Y ) ; :: thesis: ((GoB h) * (i1,I)) `1 <= (h /. i) `1

then A3: i in dom h by FINSEQ_3:25;

then consider i2, j2 being Nat such that

A4: [i2,j2] in Indices (GoB h) and

A5: h /. i = (GoB h) * (i2,j2) by GOBOARD5:11;

A6: i2 <= len (GoB h) by A4, MATRIX_0:32;

A7: 1 <= i2 by A4, MATRIX_0:32;

then A8: [i2,I] in Indices (GoB h) by A2, A6, MATRIX_0:30;

A9: j2 <= width (GoB h) by A4, MATRIX_0:32;

1 <= j2 by A4, MATRIX_0:32;

then A10: (h /. i) `1 = ((GoB h) * (i2,1)) `1 by A5, A7, A6, A9, GOBOARD5:2

.= ((GoB h) * (i2,I)) `1 by A2, A7, A6, GOBOARD5:2 ;

i1 in Y by A2, XXREAL_2:def 7;

then ex j being Element of NAT st

( i1 = j & [j,I] in Indices (GoB h) & ex k being Nat st

( k in dom h & h /. k = (GoB h) * (j,I) ) ) by A2;

then A11: 1 <= i1 by MATRIX_0:32;

A12: i2 in NAT by ORDINAL1:def 12;

(h /. i) `2 = ((GoB h) * (i2,I)) `2 by A2, A7, A6, GOBOARD5:1;

then h /. i = (GoB h) * (i2,I) by A10, A1, EUCLID:53;

then i2 in Y by A2, A3, A8, A12;

then A13: i1 <= i2 by A2, XXREAL_2:def 7;

A14: i2 <= len (GoB h) by A4, MATRIX_0:32;

for Y being non empty finite Subset of NAT st 1 <= i & i <= len h & 1 <= I & I <= width (GoB h) & Y = { j where j is Element of NAT : ( [j,I] in Indices (GoB h) & ex k being Nat st

( k in dom h & h /. k = (GoB h) * (j,I) ) ) } & (h /. i) `2 = ((GoB h) * (1,I)) `2 & i1 = min Y holds

((GoB h) * (i1,I)) `1 <= (h /. i) `1

let I, i1, i be Nat; :: thesis: for Y being non empty finite Subset of NAT st 1 <= i & i <= len h & 1 <= I & I <= width (GoB h) & Y = { j where j is Element of NAT : ( [j,I] in Indices (GoB h) & ex k being Nat st

( k in dom h & h /. k = (GoB h) * (j,I) ) ) } & (h /. i) `2 = ((GoB h) * (1,I)) `2 & i1 = min Y holds

((GoB h) * (i1,I)) `1 <= (h /. i) `1

let Y be non empty finite Subset of NAT; :: thesis: ( 1 <= i & i <= len h & 1 <= I & I <= width (GoB h) & Y = { j where j is Element of NAT : ( [j,I] in Indices (GoB h) & ex k being Nat st

( k in dom h & h /. k = (GoB h) * (j,I) ) ) } & (h /. i) `2 = ((GoB h) * (1,I)) `2 & i1 = min Y implies ((GoB h) * (i1,I)) `1 <= (h /. i) `1 )

A1: h /. i = |[((h /. i) `1),((h /. i) `2)]| by EUCLID:53;

assume A2: ( 1 <= i & i <= len h & 1 <= I & I <= width (GoB h) & Y = { j where j is Element of NAT : ( [j,I] in Indices (GoB h) & ex k being Nat st

( k in dom h & h /. k = (GoB h) * (j,I) ) ) } & (h /. i) `2 = ((GoB h) * (1,I)) `2 & i1 = min Y ) ; :: thesis: ((GoB h) * (i1,I)) `1 <= (h /. i) `1

then A3: i in dom h by FINSEQ_3:25;

then consider i2, j2 being Nat such that

A4: [i2,j2] in Indices (GoB h) and

A5: h /. i = (GoB h) * (i2,j2) by GOBOARD5:11;

A6: i2 <= len (GoB h) by A4, MATRIX_0:32;

A7: 1 <= i2 by A4, MATRIX_0:32;

then A8: [i2,I] in Indices (GoB h) by A2, A6, MATRIX_0:30;

A9: j2 <= width (GoB h) by A4, MATRIX_0:32;

1 <= j2 by A4, MATRIX_0:32;

then A10: (h /. i) `1 = ((GoB h) * (i2,1)) `1 by A5, A7, A6, A9, GOBOARD5:2

.= ((GoB h) * (i2,I)) `1 by A2, A7, A6, GOBOARD5:2 ;

i1 in Y by A2, XXREAL_2:def 7;

then ex j being Element of NAT st

( i1 = j & [j,I] in Indices (GoB h) & ex k being Nat st

( k in dom h & h /. k = (GoB h) * (j,I) ) ) by A2;

then A11: 1 <= i1 by MATRIX_0:32;

A12: i2 in NAT by ORDINAL1:def 12;

(h /. i) `2 = ((GoB h) * (i2,I)) `2 by A2, A7, A6, GOBOARD5:1;

then h /. i = (GoB h) * (i2,I) by A10, A1, EUCLID:53;

then i2 in Y by A2, A3, A8, A12;

then A13: i1 <= i2 by A2, XXREAL_2:def 7;

A14: i2 <= len (GoB h) by A4, MATRIX_0:32;

now :: thesis: ( ( i1 < i2 & ((GoB h) * (i1,I)) `1 <= ((GoB h) * (i2,I)) `1 ) or ( i1 >= i2 & ((GoB h) * (i1,I)) `1 <= ((GoB h) * (i2,I)) `1 ) )

hence
((GoB h) * (i1,I)) `1 <= (h /. i) `1
by A10; :: thesis: verumend;