let i, j be Nat; :: thesis: ex n being Nat st square-uparrow n c= [:(NAT \ (Segm i)),(NAT \ (Segm j)):]
reconsider n = max (i,j) as Element of NAT by ORDINAL1:def 12;
take n ; :: thesis: square-uparrow n c= [:(NAT \ (Segm i)),(NAT \ (Segm j)):]
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in square-uparrow n or x in [:(NAT \ (Segm i)),(NAT \ (Segm j)):] )
assume A1: x in square-uparrow n ; :: thesis: x in [:(NAT \ (Segm i)),(NAT \ (Segm j)):]
then reconsider y = x as Element of [:NAT,NAT:] ;
consider n1, n2 being Nat such that
A2: y `1 = n1 and
A3: y `2 = n2 and
A4: n <= n1 and
A5: n <= n2 by A1, Def3;
A6: y is pair by Th4;
i <= n by XXREAL_0:25;
then A7: not n1 in Segm i by A4, XXREAL_0:2, NAT_1:44;
n1 in NAT by ORDINAL1:def 12;
then A8: n1 in NAT \ (Segm i) by A7, XBOOLE_0:def 5;
j <= n by XXREAL_0:25;
then A9: not n2 in Segm j by A5, XXREAL_0:2, NAT_1:44;
n2 in NAT by ORDINAL1:def 12;
then n2 in NAT \ (Segm j) by A9, XBOOLE_0:def 5;
hence x in [:(NAT \ (Segm i)),(NAT \ (Segm j)):] by A2, A3, A6, A8, ZFMISC_1:def 2; :: thesis: verum