reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;

theorem
  p1`1 <= p2`1 & p in LSeg(p1,p2) implies p1`1 <= p`1 & p`1 <= p2`1
proof
  assume that
A1: p1`1 <= p2`1 and
A2: p in LSeg(p1,p2);
  consider lambda such that
A3: p = (1-lambda)*p1 + lambda*p2 and
A4: 0 <= lambda and
A5: lambda <= 1 by A2;
A6: ((1-lambda)*p1)`1 = |[(1-lambda)*p1`1, (1-lambda)*p1`2]|`1 by EUCLID:57
    .= (1-lambda)*p1`1 by EUCLID:52;
A7: (lambda*p2)`1 = |[lambda*p2`1, lambda*p2`2]|`1 by EUCLID:57
    .= lambda*p2`1 by EUCLID:52;
A8: p`1 = |[((1-lambda)*p1)`1 + (lambda*p2)`1, ((1-lambda)*p1)`2 + (lambda*
  p2)`2]|`1 by A3,EUCLID:55
    .= (1-lambda)*p1`1 + lambda*p2`1 by A6,A7,EUCLID:52;
  lambda*p1`1 <= lambda*p2`1 by A1,A4,XREAL_1:64;
  then (1-lambda)*p1`1 + lambda*p1`1 <= p`1 by A8,XREAL_1:7;
  hence p1`1 <= p`1;
  0 <= 1-lambda by A5,XREAL_1:48;
  then (1-lambda)*p1`1 <= (1-lambda)*p2`1 by A1,XREAL_1:64;
  then p`1 <= (1-lambda)*p2`1 + lambda*p2`1 by A8,XREAL_1:6;
  hence thesis;
end;
