let p, q, p1, q1 be Point of ; :: thesis: ( LSeg p,q is horizontal & LSeg p1,q1 is horizontal & p `2 = p1 `2 & p `1 <= p1 `1 & p1 `1 <= q1 `1 & q1 `1 <= q `1 implies LSeg p1,q1 c= LSeg p,q )
assume that
A1: LSeg p,q is horizontal and
A2: LSeg p1,q1 is horizontal and
A3: p `2 = p1 `2 and
A4: p `1 <= p1 `1 and
A5: p1 `1 <= q1 `1 and
A6: q1 `1 <= q `1 ; :: thesis: LSeg p1,q1 c= LSeg p,q
A7: p `2 = q `2 by A1, SPPOL_1:36;
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg p1,q1 or x in LSeg p,q )
assume A8: x in LSeg p1,q1 ; :: thesis: x in LSeg p,q
then reconsider r = x as Point of ;
p1 `1 <= r `1 by A5, A8, TOPREAL1:9;
then A9: p `1 <= r `1 by A4, XXREAL_0:2;
r `1 <= q1 `1 by A5, A8, TOPREAL1:9;
then A10: r `1 <= q `1 by A6, XXREAL_0:2;
p1 `2 = r `2 by A2, A8, SPPOL_1:63;
hence x in LSeg p,q by A3, A7, A9, A10, Th9; :: thesis: verum