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

theorem Th16:
  |.p3-p1.|=|.p2-p3.| & p1<>p2 implies angle(p3,p1,p2)=angle(p1,p2 ,p3)
proof
  assume
A1: |.p3-p1.|=|.p2-p3.|;
  assume
A2: p1<>p2;
  per cases;
  suppose
A3: p1,p2,p3 are_mutually_distinct;
    |.p3-p1.|^2 = |.p1-p2.|^2 + |.p3-p2.|^2 - 2*|.p1-p2.|*|.p3-p2.| * cos
    angle(p1,p2,p3) by Th7;
    then |.p1-p2.|^2 + |.p3-p2.|^2 - 2*|.p1-p2.|*|.p3-p2.| * cos angle(p1,p2,
p3) = |.p3-p1.|^2 + |.p2-p1.|^2 - 2*|.p3-p1.|*|.p2-p1.| * cos angle(p3,p1,p2)
    by A1,Th7;
    then
    |.p3-p2.|^2 - 2*|.p1-p2.|*|.p3-p2.| * cos angle(p1,p2,p3) + |.p1-p2.|
^2 = |.p3-p1.|^2 - 2*|.p3-p1.|*|.p2-p1.| * cos angle(p3,p1,p2) + |.p2-p1.|^2;
    then
    |.p3-p2.|^2 - 2*|.p1-p2.|*|.p3-p2.| * cos angle(p1,p2,p3) + |.p1-p2.|
^2 = |.p3-p1.|^2 - 2*|.p3-p1.|*|.p2-p1.| * cos angle(p3,p1,p2) + |.p1-p2.|^2
by Lm2;
    then - 2*|.p1-p2.|*|.p3-p2.| * cos angle(p1,p2,p3) + |.p3-p2.|^2 = - 2*|.
    p2-p3.|*|.p2-p1.| * cos angle(p3,p1,p2) + |.p2-p3.|^2 by A1;
    then - 2*|.p1-p2.|*|.p3-p2.| * cos angle(p1,p2,p3) + |.p3-p2.|^2 = - 2*|.
    p2-p3.|*|.p2-p1.| * cos angle(p3,p1,p2) + |.p3-p2.|^2 by Lm2;
    then
    |.p1-p2.|*|.p3-p2.| * cos angle(p1,p2,p3) = |.p2-p3.|*|.p2-p1.| * cos
    angle(p3,p1,p2);
    then
    |.p1-p2.|*|.p3-p2.| * cos angle(p1,p2,p3) = |.p2-p3.|*|.p1-p2.| * cos
    angle(p3,p1,p2) by Lm2;
    then
A4: |.p3-p2.| * cos angle(p1,p2,p3)* |.p1-p2.| = |.p2-p3.| * cos angle(p3
    ,p1,p2)* |.p1-p2.|;
    p1<>p2 by A3,ZFMISC_1:def 5;
    then |.p1-p2.|<>0 by Lm1;
    then |.p3-p2.| * cos angle(p1,p2,p3) = |.p2-p3.| * cos angle(p3,p1,p2) by
A4,XCMPLX_1:5;
    then
A5: |.p2-p3.| * cos angle(p1,p2,p3) = |.p2-p3.| * cos angle(p3,p1,p2) by Lm2;
    p1<>p3 by A3,ZFMISC_1:def 5;
    then
A6: |.p3-p1.|<>0 by Lm1;
    |.p3-p1.| * sin angle(p3,p1,p2) = |.p3-p2.| * sin angle(p1,p2,p3) by A2,Th6
      .= |.p3-p1.| * sin angle(p1,p2,p3) by A1,Lm2;
    then
A7: sin angle(p3,p1,p2) = sin angle(p1,p2,p3) by A6,XCMPLX_1:5;
    p2<>p3 by A3,ZFMISC_1:def 5;
    then |.p2-p3.|<>0 by Lm1;
    then cos angle(p1,p2,p3) = cos angle(p3,p1,p2) by A5,XCMPLX_1:5;
    hence thesis by A7,Th1;
  end;
  suppose
A8: not p1,p2,p3 are_mutually_distinct;
    per cases by A8,ZFMISC_1:def 5;
    suppose
      p1=p2;
      hence thesis by A2;
    end;
    suppose
A9:   p1=p3;
      then |.p2-p3.| = 0 by A1,Lm1;
      then p2=p3 by Lm1;
      hence thesis by A9;
    end;
    suppose
A10:  p2=p3;
      then |.p3-p1.| = 0 by A1,Lm1;
      then p3=p1 by Lm1;
      hence thesis by A10;
    end;
  end;
end;
