let n be Nat; :: thesis: for f being non empty FinSequence of (TOP-REAL 2) st n in dom f & ( for m being Nat st m in dom f holds

(X_axis f) . m <= (X_axis f) . n ) holds

f /. n in rng (Line ((GoB f),(len (GoB f))))

let f be non empty FinSequence of (TOP-REAL 2); :: thesis: ( n in dom f & ( for m being Nat st m in dom f holds

(X_axis f) . m <= (X_axis f) . n ) implies f /. n in rng (Line ((GoB f),(len (GoB f)))) )

set x = X_axis f;

set y = Y_axis f;

set r = (X_axis f) . n;

assume that

A1: n in dom f and

A2: for m being Nat st m in dom f holds

(X_axis f) . m <= (X_axis f) . n ; :: thesis: f /. n in rng (Line ((GoB f),(len (GoB f))))

reconsider p = f /. n as Point of (TOP-REAL 2) ;

A3: dom f = Seg (len f) by FINSEQ_1:def 3;

A4: ( dom (X_axis f) = Seg (len (X_axis f)) & len (X_axis f) = len f ) by FINSEQ_1:def 3, GOBOARD1:def 1;

then A5: (X_axis f) . n = p `1 by A1, A3, GOBOARD1:def 1;

A6: rng (Incr (X_axis f)) = rng (X_axis f) by SEQ_4:def 21;

(X_axis f) . n in rng (X_axis f) by A1, A3, A4, FUNCT_1:def 3;

then consider i being Nat such that

A7: i in dom (Incr (X_axis f)) and

A8: (Incr (X_axis f)) . i = p `1 by A5, A6, FINSEQ_2:10;

reconsider i = i as Element of NAT by ORDINAL1:def 12;

A9: i <= len (Incr (X_axis f)) by A7, FINSEQ_3:25;

A10: 1 <= i by A7, FINSEQ_3:25;

( dom (Y_axis f) = Seg (len (Y_axis f)) & len (Y_axis f) = len f ) by FINSEQ_1:def 3, GOBOARD1:def 2;

then ( (Y_axis f) . n = p `2 & (Y_axis f) . n in rng (Y_axis f) ) by A1, A3, FUNCT_1:def 3, GOBOARD1:def 2;

then consider j being Nat such that

A16: j in dom (Incr (Y_axis f)) and

A17: (Incr (Y_axis f)) . j = p `2 by A15, FINSEQ_2:10;

A18: p = |[(p `1),(p `2)]| by EUCLID:53;

len (Line ((GoB f),(len (GoB f)))) = width (GoB f) by MATRIX_0:def 7;

then A19: dom (Line ((GoB f),(len (GoB f)))) = Seg (width (GoB f)) by FINSEQ_1:def 3;

( width (GoB f) = card (rng (Y_axis f)) & len (Incr (Y_axis f)) = card (rng (Y_axis f)) ) by Th13, SEQ_4:def 21;

then A20: dom (Incr (Y_axis f)) = Seg (width (GoB f)) by FINSEQ_1:def 3;

A21: ( len (GoB f) = card (rng (X_axis f)) & len (Incr (X_axis f)) = card (rng (X_axis f)) ) by Th13, SEQ_4:def 21;

then ( Indices (GoB f) = [:(dom (GoB f)),(Seg (width (GoB f))):] & dom (Incr (X_axis f)) = dom (GoB f) ) by FINSEQ_3:29, MATRIX_0:def 4;

then [(len (GoB f)),j] in Indices (GoB f) by A21, A7, A16, A20, A11, ZFMISC_1:87;

then (GoB f) * ((len (GoB f)),j) = |[(p `1),(p `2)]| by A21, A8, A17, A11, Def1;

then (Line ((GoB f),(len (GoB f)))) . j = f /. n by A16, A20, A18, MATRIX_0:def 7;

hence f /. n in rng (Line ((GoB f),(len (GoB f)))) by A16, A20, A19, FUNCT_1:def 3; :: thesis: verum

(X_axis f) . m <= (X_axis f) . n ) holds

f /. n in rng (Line ((GoB f),(len (GoB f))))

let f be non empty FinSequence of (TOP-REAL 2); :: thesis: ( n in dom f & ( for m being Nat st m in dom f holds

(X_axis f) . m <= (X_axis f) . n ) implies f /. n in rng (Line ((GoB f),(len (GoB f)))) )

set x = X_axis f;

set y = Y_axis f;

set r = (X_axis f) . n;

assume that

A1: n in dom f and

A2: for m being Nat st m in dom f holds

(X_axis f) . m <= (X_axis f) . n ; :: thesis: f /. n in rng (Line ((GoB f),(len (GoB f))))

reconsider p = f /. n as Point of (TOP-REAL 2) ;

A3: dom f = Seg (len f) by FINSEQ_1:def 3;

A4: ( dom (X_axis f) = Seg (len (X_axis f)) & len (X_axis f) = len f ) by FINSEQ_1:def 3, GOBOARD1:def 1;

then A5: (X_axis f) . n = p `1 by A1, A3, GOBOARD1:def 1;

A6: rng (Incr (X_axis f)) = rng (X_axis f) by SEQ_4:def 21;

(X_axis f) . n in rng (X_axis f) by A1, A3, A4, FUNCT_1:def 3;

then consider i being Nat such that

A7: i in dom (Incr (X_axis f)) and

A8: (Incr (X_axis f)) . i = p `1 by A5, A6, FINSEQ_2:10;

reconsider i = i as Element of NAT by ORDINAL1:def 12;

A9: i <= len (Incr (X_axis f)) by A7, FINSEQ_3:25;

A10: 1 <= i by A7, FINSEQ_3:25;

A11: now :: thesis: not i <> len (Incr (X_axis f))

A15:
rng (Incr (Y_axis f)) = rng (Y_axis f)
by SEQ_4:def 21;reconsider s = (Incr (X_axis f)) . (i + 1) as Real ;

assume i <> len (Incr (X_axis f)) ; :: thesis: contradiction

then i < len (Incr (X_axis f)) by A9, XXREAL_0:1;

then A12: i + 1 <= len (Incr (X_axis f)) by NAT_1:13;

1 <= i + 1 by A10, NAT_1:13;

then A13: i + 1 in dom (Incr (X_axis f)) by A12, FINSEQ_3:25;

then (Incr (X_axis f)) . (i + 1) in rng (Incr (X_axis f)) by FUNCT_1:def 3;

then A14: ex m being Nat st

( m in dom (X_axis f) & (X_axis f) . m = s ) by A6, FINSEQ_2:10;

i < i + 1 by NAT_1:13;

then (X_axis f) . n < s by A5, A7, A8, A13, SEQM_3:def 1;

hence contradiction by A2, A3, A4, A14; :: thesis: verum

end;assume i <> len (Incr (X_axis f)) ; :: thesis: contradiction

then i < len (Incr (X_axis f)) by A9, XXREAL_0:1;

then A12: i + 1 <= len (Incr (X_axis f)) by NAT_1:13;

1 <= i + 1 by A10, NAT_1:13;

then A13: i + 1 in dom (Incr (X_axis f)) by A12, FINSEQ_3:25;

then (Incr (X_axis f)) . (i + 1) in rng (Incr (X_axis f)) by FUNCT_1:def 3;

then A14: ex m being Nat st

( m in dom (X_axis f) & (X_axis f) . m = s ) by A6, FINSEQ_2:10;

i < i + 1 by NAT_1:13;

then (X_axis f) . n < s by A5, A7, A8, A13, SEQM_3:def 1;

hence contradiction by A2, A3, A4, A14; :: thesis: verum

( dom (Y_axis f) = Seg (len (Y_axis f)) & len (Y_axis f) = len f ) by FINSEQ_1:def 3, GOBOARD1:def 2;

then ( (Y_axis f) . n = p `2 & (Y_axis f) . n in rng (Y_axis f) ) by A1, A3, FUNCT_1:def 3, GOBOARD1:def 2;

then consider j being Nat such that

A16: j in dom (Incr (Y_axis f)) and

A17: (Incr (Y_axis f)) . j = p `2 by A15, FINSEQ_2:10;

A18: p = |[(p `1),(p `2)]| by EUCLID:53;

len (Line ((GoB f),(len (GoB f)))) = width (GoB f) by MATRIX_0:def 7;

then A19: dom (Line ((GoB f),(len (GoB f)))) = Seg (width (GoB f)) by FINSEQ_1:def 3;

( width (GoB f) = card (rng (Y_axis f)) & len (Incr (Y_axis f)) = card (rng (Y_axis f)) ) by Th13, SEQ_4:def 21;

then A20: dom (Incr (Y_axis f)) = Seg (width (GoB f)) by FINSEQ_1:def 3;

A21: ( len (GoB f) = card (rng (X_axis f)) & len (Incr (X_axis f)) = card (rng (X_axis f)) ) by Th13, SEQ_4:def 21;

then ( Indices (GoB f) = [:(dom (GoB f)),(Seg (width (GoB f))):] & dom (Incr (X_axis f)) = dom (GoB f) ) by FINSEQ_3:29, MATRIX_0:def 4;

then [(len (GoB f)),j] in Indices (GoB f) by A21, A7, A16, A20, A11, ZFMISC_1:87;

then (GoB f) * ((len (GoB f)),j) = |[(p `1),(p `2)]| by A21, A8, A17, A11, Def1;

then (Line ((GoB f),(len (GoB f)))) . j = f /. n by A16, A20, A18, MATRIX_0:def 7;

hence f /. n in rng (Line ((GoB f),(len (GoB f)))) by A16, A20, A19, FUNCT_1:def 3; :: thesis: verum