let a, b, c, d be Real; :: thesis: ( a <= b & c <= d implies N-bound (rectangle (a,b,c,d)) = d )
assume that
A1: a <= b and
A2: c <= d ; :: thesis: N-bound (rectangle (a,b,c,d)) = d
set X = rectangle (a,b,c,d);
reconsider Z = (proj2 | (rectangle (a,b,c,d))) .: the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) as Subset of REAL ;
A3: rectangle (a,b,c,d) = the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) by PRE_TOPC:8;
A4: for p being Real st p in Z holds
p <= d

proof

let p be Real; :: thesis: ( p in Z implies p <= d )
assume p in Z ; :: thesis: p <= d
then consider p0 being object such that
A5: p0 in the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) and
p0 in the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) and
A6: p = (proj2 | (rectangle (a,b,c,d))) . p0 by FUNCT_2:64;
reconsider p0 = p0 as Point of (TOP-REAL 2) by A3, A5;
rectangle (a,b,c,d) = { q where q is Point of (TOP-REAL 2) : ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) } by A1, A2, SPPOL_2:54;
then ex q being Point of (TOP-REAL 2) st
( p0 = q & ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) ) by A3, A5;
hence p <= d by A2, A3, A5, A6, PSCOMP_1:23; :: thesis: verum

end;

A7: for q being Real st ( for p being Real st p in Z holds
p <= q ) holds
d <= q

proof

let q be Real; :: thesis: ( ( for p being Real st p in Z holds
p <= q ) implies d <= q )
assume A8: for p being Real st p in Z holds
p <= q ; :: thesis: d <= q
|[b,d]| in LSeg (|[b,c]|,|[b,d]|) by RLTOPSP1:68;
then A9: |[b,d]| in (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by XBOOLE_0:def 3;
rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def 3;
then A10: |[b,d]| in rectangle (a,b,c,d) by A9, XBOOLE_0:def 3;
then (proj2 | (rectangle (a,b,c,d))) . |[b,d]| = |[b,d]| `2 by PSCOMP_1:23
.= d by EUCLID:52 ;
hence d <= q by A3, A8, A10, FUNCT_2:35; :: thesis: verum

end;

thus N-bound (rectangle (a,b,c,d)) = upper_bound (proj2 | (rectangle (a,b,c,d))) by PSCOMP_1:def 8
.= upper_bound Z by PSCOMP_1:def 2
.= d by A4, A7, SEQ_4:46 ; :: thesis: verum