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

theorem Th25:
  p in LSeg(p1,p2) & p1,p2,p3 is_a_triangle & angle(p1,p3,p2) =
  angle(p,p3,p2) implies p=p1
proof
  assume
A1: p in LSeg(p1,p2);
  assume
A2: p1,p2,p3 is_a_triangle;
  then
A3: angle(p3,p1,p2)<>PI by Th20;
A4: p1,p2,p3 are_mutually_distinct by A2,Th20;
  then p1<>p2 by ZFMISC_1:def 5;
  then
A5: euc2cpx(p2)<> euc2cpx(p1) by EUCLID_3:4;
A6: not p3 in LSeg(p1,p2) by A2,TOPREAL9:67;
  ( not p1 in LSeg(p2,p3))& not p2 in LSeg(p3,p1) by A2,TOPREAL9:67;
  then
A7: p1,p3,p2 is_a_triangle by A6,TOPREAL9:67;
  p2<>p3 by A4,ZFMISC_1:def 5;
  then
A8: |.p2-p3.|<>0 by Lm1;
A9: p2<>p3 by A4,ZFMISC_1:def 5;
  then
A10: euc2cpx(p3)<> euc2cpx(p2) by EUCLID_3:4;
  assume
A11: angle(p1,p3,p2) = angle(p,p3,p2);
  angle(p2,p3,p1)<>PI by A2,Th20;
  then
A12: angle(p,p3,p2)<>PI by A11,COMPLEX2:82;
A13: p<>p3 by A1,A2,TOPREAL9:67;
  then
A14: euc2cpx(p)<> euc2cpx(p3) by EUCLID_3:4;
  p1<>p3 by A4,ZFMISC_1:def 5;
  then
A15: euc2cpx(p3)<> euc2cpx(p1) by EUCLID_3:4;
A16: angle(p1,p2,p3)<>PI by A2,Th20;
A17: p<>p2
  proof
    assume p=p2;
    then angle(p1,p3,p2)=0 by A11,COMPLEX2:79;
    then angle(p3,p2,p1) = 0 & angle(p2,p1,p3) = PI or angle(p3,p2,p1) = PI &
    angle(p2,p1,p3) = 0 by A10,A15,A5,COMPLEX2:87;
    hence contradiction by A16,A3,COMPLEX2:82;
  end;
  then
A18: angle(p3,p2,p1) = angle(p3,p2,p) by A1,Th10;
  then
A19: angle(p3,p2,p)<>PI by A16,COMPLEX2:82;
A20: p,p3,p2 are_mutually_distinct by A9,A17,A13,ZFMISC_1:def 5;
A21: euc2cpx(p)<> euc2cpx(p2) by A17,EUCLID_3:4;
A22: angle(p2,p1,p3) = angle(p2,p,p3)
  proof
    per cases by A10,A15,A5,A14,A21,COMPLEX2:88;
    suppose
      angle(p1,p3,p2)+angle(p3,p2,p1)+angle(p2,p1,p3) = PI & angle(p,
      p3,p2)+angle(p3,p2,p)+angle(p2,p,p3) = PI;
      hence thesis by A11,A18;
    end;
    suppose
      angle(p1,p3,p2)+angle(p3,p2,p1)+angle(p2,p1,p3) = 5*PI & angle(
      p,p3,p2)+angle(p3,p2,p)+angle(p2,p,p3) = 5*PI;
      hence thesis by A11,A18;
    end;
    suppose
A23:  angle(p1,p3,p2)+angle(p3,p2,p1)+angle(p2,p1,p3) = PI & angle(p,
      p3,p2)+angle(p3,p2,p)+angle(p2,p,p3) = 5*PI;
      angle(p2,p1,p3)>=0 & -angle(p2,p,p3) > -2*PI by COMPLEX2:70,XREAL_1:24;
      then
A24:  angle(p2,p1,p3)+(-angle(p2,p,p3)) > 0+(-2*PI) by XREAL_1:8;
      angle(p2,p1,p3)-angle(p2,p,p3) = -4*PI by A11,A18,A23;
      then 4*PI < 2*PI by A24,XREAL_1:24;
      then 4*PI/PI < 2*PI/PI by XREAL_1:74;
      then 4 < 2*PI/PI by XCMPLX_1:89;
      then 4 < 2 by XCMPLX_1:89;
      hence thesis;
    end;
    suppose
A25:  angle(p1,p3,p2)+angle(p3,p2,p1)+angle(p2,p1,p3)=5*PI & angle(p,
      p3,p2)+angle(p3,p2,p)+angle(p2,p,p3)=PI;
      angle(p2,p1,p3)<2*PI & angle(p2,p,p3)>=0 by COMPLEX2:70;
      then angle(p2,p1,p3)+(-angle(p2,p,p3)) < 2*PI+(-0) by XREAL_1:8;
      then 4*PI/PI < 2*PI/PI by A11,A18,A25,XREAL_1:74;
      then 4 < 2*PI/PI by XCMPLX_1:89;
      then 4 < 2 by XCMPLX_1:89;
      hence thesis;
    end;
  end;
  then angle(p2,p,p3)<>PI by A3,COMPLEX2:82;
  then p,p3,p2 is_a_triangle by A20,A12,A19,Th20;
  then |.p2-p3.|*|.p-p2.| = |.p1-p2.|*|.p2-p3.| by A7,A11,A22,Th21;
  then |.p-p2.| = |.p1-p2.| by A8,XCMPLX_1:5;
  then
A26: |.p2-p.| = |.p1-p2.| by Lm2
    .= |.p2-p1.| by Lm2;
  assume
A27: p1<>p;
A28: |.p2-p1.|^2 = |.p1-p.|^2 + |.p2-p.|^2 -2*|.p1-p.|*|.p2-p.|*cos angle(p1
  ,p,p2) by Th7
    .= |.p1-p.|^2 + |.p2-p.|^2 -2*|.p1-p.|*|.p2-p.|*(-1) by A1,A27,A17,Th8,
SIN_COS:77;
  per cases by A26,A28;
  suppose
    |.p1-p.|=0;
    hence contradiction by A27,Lm1;
  end;
  suppose
    |.p1-p.| + 2*|.p2-p.|=0;
    then |.p1-p.|=0;
    hence contradiction by A27,Lm1;
  end;
end;
