reserve p1,p2,p3,p4,p5,p6,p,pc for Point of TOP-REAL 2;
reserve a,b,c,r,s for Real;

theorem Th17:
  for p1,p2,p3,p st p in LSeg(p1,p2) & p<>p2 holds |(p3-p,p2-p1)|
  = 0 iff |(p3-p,p2-p)| = 0
proof
  let p1,p2,p3,p;
  assume p in LSeg(p1,p2);
  then p in LSeg(p2,p1);
  then consider l be Real such that
A1: p=(1-l)*p2+l*p1 and
  0<=l and
  l<=1;
  assume
A2: p<>p2;
A3: p2-p = p2-((1-l)*p2)-l*p1 by A1,RLVECT_1:27
    .= p2-(1*p2-l*p2)-l*p1 by RLVECT_1:35
    .= p2-(p2-l*p2)-l*p1 by RLVECT_1:def 8
    .= p2-p2+l*p2-l*p1 by RLVECT_1:29
    .= 0.TOP-REAL 2+l*p2-l*p1 by RLVECT_1:5
    .= l*p2-l*p1 by RLVECT_1:4
    .= l*(p2-p1) by RLVECT_1:34;
  hereby
    assume
A4: |(p3-p,p2-p1)| = 0;
    thus |(p3-p,p2-p)| = l*|(p3-p,p2-p1)| by A3,EUCLID_2:20
      .= 0 by A4;
  end;
  assume |(p3-p,p2-p)| = 0;
  then
A5: l*|(p3-p,p2-p1)| = 0 by A3,EUCLID_2:20;
  per cases by A5;
  suppose
    l=0;
    then p = 1*p2+0.TOP-REAL 2 by A1,RLVECT_1:10
      .= 1*p2 by RLVECT_1:4
      .= p2 by RLVECT_1:def 8;
    hence thesis by A2;
  end;
  suppose
    |(p3-p,p2-p1)| = 0;
    hence thesis;
  end;
end;
