let a, b, c, d be real number ; :: thesis: ( a <= b & c <= d implies E-most (rectangle a,b,c,d) = LSeg |[b,c]|,|[b,d]| )
set K = rectangle a,b,c,d;
assume A1: ( a <= b & c <= d ) ; :: thesis: E-most (rectangle a,b,c,d) = LSeg |[b,c]|,|[b,d]|
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 A2: (LSeg |[a,c]|,|[b,c]|) \/ (LSeg |[b,c]|,|[b,d]|) c= rectangle a,b,c,d by XBOOLE_1:7;
A3: LSeg |[b,c]|,|[b,d]| c= (LSeg |[a,c]|,|[b,c]|) \/ (LSeg |[b,c]|,|[b,d]|) by XBOOLE_1:7;
A4: SE-corner (rectangle a,b,c,d) = |[(E-bound (rectangle a,b,c,d)),(S-bound (rectangle a,b,c,d))]| by PSCOMP_1:def 37;
A5: NE-corner (rectangle a,b,c,d) = |[b,d]| by A1, Th51;
A6: E-bound (rectangle a,b,c,d) = b by A1, Th48;
A7: S-bound (rectangle a,b,c,d) = c by A1, Th49;
thus E-most (rectangle a,b,c,d) = (LSeg (SE-corner (rectangle a,b,c,d)),(NE-corner (rectangle a,b,c,d))) /\ (rectangle a,b,c,d) by PSCOMP_1:def 40
.= LSeg |[b,c]|,|[b,d]| by A2, A3, A4, A5, A6, A7, XBOOLE_1:1, XBOOLE_1:28 ; :: thesis: verum