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 Th52:
  p`1 <= q`1 implies proj1.:LSeg(p,q) = [.p`1,q`1.]
proof
  assume
A1: p`1 <= q`1;
  for y being object holds y in [.p`1,q`1.] iff
   ex x being object st x in dom
  proj1 & x in LSeg(p,q) & y = proj1.x
  proof
    let y be object;
    hereby
      assume
A2:   y in [.p`1,q`1.];
      then reconsider r = y as Real;
      consider t such that
A3:   0 <=t and
A4:   t <= 1 and
A5:   r = t*p`1 + (1-t)*q`1 by A1,A2,Th51;
      set o = t*p + (1-t)*q;
      reconsider x = o as object;
      take x;
      o in the carrier of TOP-REAL 2;
      hence x in dom proj1 by FUNCT_2:def 1;
      o in LSeg(q,p) by A3,A4;
      hence x in LSeg(p,q);
      thus y = (t*p)`1 + (1-t)*(q`1) by A5,TOPREAL3:4
        .= (t*p)`1 + ((1-t)*q)`1 by TOPREAL3:4
        .= (t*p + (1-t)*q)`1 by TOPREAL3:2
        .= proj1.x by PSCOMP_1:def 5;
    end;
    given x being object such that
    x in dom proj1 and
A6: x in LSeg(p,q) and
A7: y = proj1.x;
    reconsider s = x as Point of TOP-REAL 2 by A6;
    x in LSeg(q,p) by A6;
    then consider r being Real such that
A8: s = (1-r)*q+r*p and
A9: 0<=r and
A10: r<=1;
    y = s`1 by A7,PSCOMP_1:def 5
      .= ((1-r)*q)`1+(r*p)`1 by A8,TOPREAL3:2
      .= (1-r)*(q`1)+(r*p)`1 by TOPREAL3:4
      .= (1-r)*(q`1)+r*(p`1) by TOPREAL3:4;
    hence thesis by A1,A9,A10,Th51;
  end;
  hence thesis by FUNCT_1:def 6;
end;
