reserve S for Subset of TOP-REAL 2,
  C,C1,C2 for non empty compact Subset of TOP-REAL 2,
  p,q for Point of TOP-REAL 2;
reserve i,j,k for Nat,
  t,r1,r2,s1,s2 for Real;
reserve D1 for non vertical non empty compact Subset of TOP-REAL 2,
  D2 for non horizontal non empty compact Subset of TOP-REAL 2,
  D for non vertical non horizontal non empty compact Subset of TOP-REAL 2;

theorem Th54:
  p`1 <= q`1 implies W-bound LSeg(p,q) = p`1
proof
  assume
A1: p`1 <= q`1;
  then
A2: proj1.:LSeg(p,q) = [.p`1,q`1.] by Th52;
  thus W-bound LSeg(p,q) = lower_bound(proj1.:LSeg(p,q)) by Th43
    .= p`1 by A1,A2,JORDAN5A:19;
end;
