let a, b, c, d be Real; :: 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 that
A1: a <= b and
A2: 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 A3: (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) c= rectangle (a,b,c,d) by XBOOLE_1:7;
A4: LSeg (|[b,c]|,|[b,d]|) c= (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by XBOOLE_1:7;
A5: 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 14;
A6: NE-corner (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th41;
A7: E-bound (rectangle (a,b,c,d)) = b by A1, A2, Th38;
A8: S-bound (rectangle (a,b,c,d)) = c by A1, A2, Th39;
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 17
.= LSeg (|[b,c]|,|[b,d]|) by ; :: thesis: verum