reserve n,i,k,m for Nat;
reserve r,r1,r2,s,s1,s2 for Real;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL 2,
  f,f1,f2 for FinSequence of the carrier of TOP-REAL 2,
  p,p1,p2,p3,q,q3 for Point of TOP-REAL 2;

theorem Th14:
  for p1,p2 st p1`1<>p2`1 & p1`2<>p2`2 ex p st p in LSeg(p1,p2) &
  p`1<>p1`1 & p`1<>p2`1 & p`2<>p1`2 & p`2<>p2`2
proof
  let p1,p2;
  assume that
A1: p1`1<>p2`1 and
A2: p1`2<>p2`2;
  take p = (1/2)*(p1+p2);
A3: p = (1-1/2)*p1+(1/2)*p2 by RLVECT_1:def 5;
  hence p in LSeg(p1,p2);
  hereby
    assume
A4: p`1=p1`1;
    p`1 = ((1-1/2)*p1)`1+((1/2)*p2)`1 by A3,TOPREAL3:2
      .= (1-1/2)*(p1`1)+((1/2)*p2)`1 by TOPREAL3:4
      .= (1-1/2)*(p`1)+(1/2)*(p2`1) by A4,TOPREAL3:4;
    hence contradiction by A1,A4;
  end;
  hereby
    assume
A5: p`1=p2`1;
    p`1 = ((1-1/2)*p1)`1+((1/2)*p2)`1 by A3,TOPREAL3:2
      .= (1-1/2)*(p1`1)+((1/2)*p2)`1 by TOPREAL3:4
      .= (1-1/2)*(p1`1)+(1/2)*(p`1) by A5,TOPREAL3:4;
    hence contradiction by A1,A5;
  end;
  hereby
    assume
A6: p`2=p1`2;
    p`2 = ((1-1/2)*p1)`2+((1/2)*p2)`2 by A3,TOPREAL3:2
      .= (1-1/2)*(p1`2)+((1/2)*p2)`2 by TOPREAL3:4
      .= (1-1/2)*(p`2)+(1/2)*(p2`2) by A6,TOPREAL3:4;
    hence contradiction by A2,A6;
  end;
  hereby
    assume
A7: p`2=p2`2;
    p`2 = ((1-1/2)*p1)`2+((1/2)*p2)`2 by A3,TOPREAL3:2
      .= (1-1/2)*(p1`2)+((1/2)*p2)`2 by TOPREAL3:4
      .= (1-1/2)*(p1`2)+(1/2)*(p`2) by A7,TOPREAL3:4;
    hence contradiction by A2,A7;
  end;
end;
