let p, q, p1, q1 be Point of (TOP-REAL 2); :: 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_1:15;

let x be object ; :: 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 (TOP-REAL 2) ;

p1 `1 <= r `1 by A5, A8, TOPREAL1:3;

then A9: p `1 <= r `1 by A4, XXREAL_0:2;

r `1 <= q1 `1 by A5, A8, TOPREAL1:3;

then A10: r `1 <= q `1 by A6, XXREAL_0:2;

p1 `2 = r `2 by A2, A8, SPPOL_1:40;

hence x in LSeg (p,q) by A3, A7, A9, A10, Th8; :: thesis: verum

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_1:15;

let x be object ; :: 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 (TOP-REAL 2) ;

p1 `1 <= r `1 by A5, A8, TOPREAL1:3;

then A9: p `1 <= r `1 by A4, XXREAL_0:2;

r `1 <= q1 `1 by A5, A8, TOPREAL1:3;

then A10: r `1 <= q `1 by A6, XXREAL_0:2;

p1 `2 = r `2 by A2, A8, SPPOL_1:40;

hence x in LSeg (p,q) by A3, A7, A9, A10, Th8; :: thesis: verum