let p, q, p1, q1 be Point of (TOP-REAL 2); :: thesis: ( LSeg p,q is vertical & LSeg p1,q1 is vertical & p `1 = p1 `1 & p `2 <= p1 `2 & p1 `2 <= q1 `2 & q1 `2 <= q `2 implies LSeg p1,q1 c= LSeg p,q )
assume that
A1: LSeg p,q is vertical and
A2: LSeg p1,q1 is vertical and
A3: p `1 = p1 `1 and
A4: ( p `2 <= p1 `2 & p1 `2 <= q1 `2 & q1 `2 <= q `2 ) ; :: thesis: LSeg p1,q1 c= LSeg p,q
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg p1,q1 or x in LSeg p,q )
assume A5: x in LSeg p1,q1 ; :: thesis: x in LSeg p,q
then reconsider r = x as Point of (TOP-REAL 2) ;
A6: p1 `1 = r `1 by A2, A5, SPPOL_1:64;
A7: p `1 = q `1 by A1, SPPOL_1:37;
( p1 `2 <= r `2 & r `2 <= q1 `2 ) by A4, A5, TOPREAL1:10;
then ( p `2 <= r `2 & r `2 <= q `2 ) by A4, XXREAL_0:2;
hence x in LSeg p,q by A3, A6, A7, Th8; :: thesis: verum