let p1, p2 be Point of (TOP-REAL 2); :: thesis: for b, c, d being real number st p1 `1 < b & p1 `1 = p2 `1 & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d holds
LE p1,p2, rectangle (p1 `1 ),b,c,d
let b, c, d be real number ; :: thesis: ( p1 `1 < b & p1 `1 = p2 `1 & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d implies LE p1,p2, rectangle (p1 `1 ),b,c,d )
set K = rectangle (p1 `1 ),b,c,d;
set a = p1 `1 ;
assume A1: ( p1 `1 < b & p1 `1 = p2 `1 & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d ) ; :: thesis: LE p1,p2, rectangle (p1 `1 ),b,c,d
then A2: ( c <= p1 `2 & p1 `2 < d ) by XXREAL_0:2;
then A3: c < d by XXREAL_0:2;
( c <= p2 `2 & p2 `2 <= d ) by A1, XXREAL_0:2;
then ( p1 in LSeg |[(p1 `1 ),c]|,|[(p1 `1 ),d]| & p2 in LSeg |[(p1 `1 ),c]|,|[(p1 `1 ),d]| ) by A1, A2, A3, JGRAPH_6:10;
hence LE p1,p2, rectangle (p1 `1 ),b,c,d by A1, A3, JGRAPH_6:65; :: thesis: verum