let n be Nat; 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
(Y_axis f) . n <= (Y_axis f) . m ) holds
f /. n in rng (Col ((GoB f),1))
let f be non empty FinSequence of (TOP-REAL 2); ( n in dom f & ( for m being Nat st m in dom f holds
(Y_axis f) . n <= (Y_axis f) . m ) implies f /. n in rng (Col ((GoB f),1)) )
set x = X_axis f;
set y = Y_axis f;
set r = (Y_axis f) . n;
assume that
A1:
n in dom f
and
A2:
for m being Nat st m in dom f holds
(Y_axis f) . n <= (Y_axis f) . m
; f /. n in rng (Col ((GoB f),1))
reconsider p = f /. n as Point of (TOP-REAL 2) ;
A3:
dom f = Seg (len f)
by FINSEQ_1:def 3;
A4:
( dom (Y_axis f) = Seg (len (Y_axis f)) & len (Y_axis f) = len f )
by FINSEQ_1:def 3, GOBOARD1:def 2;
then A5:
(Y_axis f) . n = p `2
by A1, A3, GOBOARD1:def 2;
A6:
rng (Incr (Y_axis f)) = rng (Y_axis f)
by SEQ_4:def 21;
(Y_axis f) . n in rng (Y_axis f)
by A1, A3, A4, FUNCT_1:def 3;
then consider j being Nat such that
A7:
j in dom (Incr (Y_axis f))
and
A8:
(Incr (Y_axis f)) . j = p `2
by A5, A6, FINSEQ_2:10;
reconsider j = j as Element of NAT by ORDINAL1:def 12;
A9:
1 <= j
by A7, FINSEQ_3:25;
then reconsider j1 = j - 1 as Element of NAT by INT_1:5;
A10:
j <= len (Incr (Y_axis f))
by A7, FINSEQ_3:25;
A11:
now not j <> 1reconsider s =
(Incr (Y_axis f)) . j1 as
Real ;
assume
j <> 1
;
contradictionthen
1
< j
by A9, XXREAL_0:1;
then
1
+ 1
<= j
by NAT_1:13;
then A12:
1
<= j1
by XREAL_1:19;
j1 <= j
by XREAL_1:44;
then
j1 <= len (Incr (Y_axis f))
by A10, XXREAL_0:2;
then A13:
j1 in dom (Incr (Y_axis f))
by A12, FINSEQ_3:25;
then
(Incr (Y_axis f)) . j1 in rng (Incr (Y_axis f))
by FUNCT_1:def 3;
then A14:
ex
m being
Nat st
(
m in dom (Y_axis f) &
(Y_axis f) . m = s )
by A6, FINSEQ_2:10;
j1 < j
by XREAL_1:44;
then
s < (Y_axis f) . n
by A5, A7, A8, A13, SEQM_3:def 1;
hence
contradiction
by A2, A3, A4, A14;
verum end;
A15:
rng (Incr (X_axis f)) = rng (X_axis f)
by SEQ_4:def 21;
( dom (X_axis f) = Seg (len (X_axis f)) & len (X_axis f) = len f )
by FINSEQ_1:def 3, GOBOARD1:def 1;
then
( (X_axis f) . n = p `1 & (X_axis f) . n in rng (X_axis f) )
by A1, A3, FUNCT_1:def 3, GOBOARD1:def 1;
then consider i being Nat such that
A16:
i in dom (Incr (X_axis f))
and
A17:
(Incr (X_axis f)) . i = p `1
by A15, FINSEQ_2:10;
A18:
p = |[(p `1),(p `2)]|
by EUCLID:53;
len (Col ((GoB f),1)) = len (GoB f)
by MATRIX_0:def 8;
then A19:
dom (Col ((GoB f),1)) = dom (GoB f)
by FINSEQ_3:29;
( 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 A20:
dom (Incr (X_axis f)) = dom (GoB f)
by FINSEQ_3:29;
( 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
( Indices (GoB f) = [:(dom (GoB f)),(Seg (width (GoB f))):] & dom (Incr (Y_axis f)) = Seg (width (GoB f)) )
by FINSEQ_1:def 3, MATRIX_0:def 4;
then
[i,1] in Indices (GoB f)
by A16, A7, A20, A11, ZFMISC_1:87;
then
(GoB f) * (i,1) = |[(p `1),(p `2)]|
by A17, A8, A11, Def1;
then
(Col ((GoB f),1)) . i = f /. n
by A16, A20, A18, MATRIX_0:def 8;
hence
f /. n in rng (Col ((GoB f),1))
by A16, A20, A19, FUNCT_1:def 3; verum