let m, n be Nat; :: thesis: ( m = n - 1 implies square-uparrow n c= [:NAT,NAT:] \ [:(Seg m),(Seg m):] )
assume A1: m = n - 1 ; :: thesis: square-uparrow n c= [:NAT,NAT:] \ [:(Seg m),(Seg m):]
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in square-uparrow n or x in [:NAT,NAT:] \ [:(Seg m),(Seg m):] )
assume A2: x in square-uparrow n ; :: thesis: x in [:NAT,NAT:] \ [:(Seg m),(Seg m):]
then reconsider y = x as Element of [:NAT,NAT:] ;
consider n1, n2 being Nat such that
A3: n1 = y `1 and
A4: n2 = y `2 and
A5: n <= n1 and
n <= n2 by A2, Def3;
not x in [:(Seg m),(Seg m):]

proof

assume x in [:(Seg m),(Seg m):] ; :: thesis: contradiction
then ex x1, x2 being object st
( x1 in Seg m & x2 in Seg m & x = [x1,x2] ) by ZFMISC_1:def 2;
then ( n1 <= m & n2 <= m ) by A3, A4, FINSEQ_1:1;
then n <= m by A5, XXREAL_0:2;
then n - n <= (n - 1) - n by A1, XREAL_1:9;
then 0 <= - 1 ;
hence contradiction ; :: thesis: verum

end;

hence x in [:NAT,NAT:] \ [:(Seg m),(Seg m):] by A2, XBOOLE_0:def 5; :: thesis: verum