let n be Nat; :: thesis: for f being non empty FinSequence of () st n in dom f & ( for m being Nat st m in dom f holds
() . n <= () . m ) holds
f /. n in rng (Line ((GoB f),1))

let f be non empty FinSequence of (); :: thesis: ( n in dom f & ( for m being Nat st m in dom f holds
() . n <= () . m ) implies f /. n in rng (Line ((GoB f),1)) )

set x = X_axis f;
set y = Y_axis f;
set r = () . n;
assume that
A1: n in dom f and
A2: for m being Nat st m in dom f holds
() . n <= () . m ; :: thesis: f /. n in rng (Line ((GoB f),1))
reconsider p = f /. n as Point of () ;
A3: dom f = Seg (len f) by FINSEQ_1:def 3;
A4: ( dom () = Seg (len ()) & len () = len f ) by ;
then A5: (X_axis f) . n = p `1 by ;
A6: rng (Incr ()) = rng () by SEQ_4:def 21;
(X_axis f) . n in rng () by ;
then consider i being Nat such that
A7: i in dom (Incr ()) and
A8: (Incr ()) . i = p `1 by ;
reconsider i = i as Element of NAT by ORDINAL1:def 12;
A9: 1 <= i by ;
then reconsider i1 = i - 1 as Element of NAT by INT_1:5;
A10: i <= len (Incr ()) by ;
A11: now :: thesis: not i <> 1
reconsider s = (Incr ()) . i1 as Real ;
assume i <> 1 ; :: thesis: contradiction
then 1 < i by ;
then 1 + 1 <= i by NAT_1:13;
then A12: 1 <= i1 by XREAL_1:19;
i1 <= i by XREAL_1:44;
then i1 <= len (Incr ()) by ;
then A13: i1 in dom (Incr ()) by ;
then (Incr ()) . i1 in rng (Incr ()) by FUNCT_1:def 3;
then A14: ex m being Nat st
( m in dom () & () . m = s ) by ;
i1 < i by XREAL_1:44;
then s < () . n by ;
hence contradiction by A2, A3, A4, A14; :: thesis: verum
end;
A15: rng (Incr ()) = rng () by SEQ_4:def 21;
( dom () = Seg (len ()) & len () = len f ) by ;
then ( (Y_axis f) . n = p `2 & () . n in rng () ) by ;
then consider j being Nat such that
A16: j in dom (Incr ()) and
A17: (Incr ()) . j = p `2 by ;
A18: p = |[(p `1),(p `2)]| by EUCLID:53;
len (Line ((GoB f),1)) = width (GoB f) by MATRIX_0:def 7;
then A19: dom (Line ((GoB f),1)) = Seg (width (GoB f)) by FINSEQ_1:def 3;
( width (GoB f) = card (rng ()) & len (Incr ()) = card (rng ()) ) by ;
then A20: dom (Incr ()) = Seg (width (GoB f)) by FINSEQ_1:def 3;
( len (GoB f) = card (rng ()) & len (Incr ()) = card (rng ()) ) by ;
then ( Indices (GoB f) = [:(dom (GoB f)),(Seg (width (GoB f))):] & dom (Incr ()) = dom (GoB f) ) by ;
then [1,j] in Indices (GoB f) by ;
then (GoB f) * (1,j) = |[(p `1),(p `2)]| by A8, A17, A11, Def1;
then (Line ((GoB f),1)) . j = f /. n by ;
hence f /. n in rng (Line ((GoB f),1)) by ; :: thesis: verum