let n be Nat; :: thesis: square-uparrow n c= [:NAT,NAT:] \ [:(Segm n),(Segm n):]

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in square-uparrow n or x in [:NAT,NAT:] \ [:(Segm n),(Segm n):] )

assume A1: x in square-uparrow n ; :: thesis: x in [:NAT,NAT:] \ [:(Segm n),(Segm n):]

then reconsider y = x as Element of [:NAT,NAT:] ;

consider n1, n2 being Nat such that

A2: n1 = y `1 and

A3: n2 = y `2 and

A4: n <= n1 and

n <= n2 by A1, Def3;

not x in [:(Segm n),(Segm n):]

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in square-uparrow n or x in [:NAT,NAT:] \ [:(Segm n),(Segm n):] )

assume A1: x in square-uparrow n ; :: thesis: x in [:NAT,NAT:] \ [:(Segm n),(Segm n):]

then reconsider y = x as Element of [:NAT,NAT:] ;

consider n1, n2 being Nat such that

A2: n1 = y `1 and

A3: n2 = y `2 and

A4: n <= n1 and

n <= n2 by A1, Def3;

not x in [:(Segm n),(Segm n):]

proof

hence
x in [:NAT,NAT:] \ [:(Segm n),(Segm n):]
by A1, XBOOLE_0:def 5; :: thesis: verum
assume
x in [:(Segm n),(Segm n):]
; :: thesis: contradiction

then ex x1, x2 being object st

( x1 in Segm n & x2 in Segm n & x = [x1,x2] ) by ZFMISC_1:def 2;

then ( n1 <= n - 1 & n2 <= n - 1 ) by A2, A3, STIRL2_1:10;

then n <= n - 1 by A4, XXREAL_0:2;

then n - n <= (n - 1) - n by XREAL_1:9;

then 0 <= - 1 ;

hence contradiction ; :: thesis: verum

end;then ex x1, x2 being object st

( x1 in Segm n & x2 in Segm n & x = [x1,x2] ) by ZFMISC_1:def 2;

then ( n1 <= n - 1 & n2 <= n - 1 ) by A2, A3, STIRL2_1:10;

then n <= n - 1 by A4, XXREAL_0:2;

then n - n <= (n - 1) - n by XREAL_1:9;

then 0 <= - 1 ;

hence contradiction ; :: thesis: verum