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

theorem Th11:
  angle(p1,p,p2) = PI implies p in LSeg(p1,p2)
proof
  assume
A1: angle(p1,p,p2) = PI;
  set r = |.p-p1.|;
  set b = p1`2;
  set a = p1`1;
A2: p1 = |[a,b]| by EUCLID:53;
  per cases;
  suppose
    p=p1 or p=p2;
    hence thesis by RLTOPSP1:68;
  end;
  suppose
A3: p<>p1 & p<>p2;
A4: |.p2-p1.|^2 = |.p1-p.|^2 + |.p2-p.|^2 - 2*|.p1-p.|*|.p2-p.|* (-1) by A1,Th7
,SIN_COS:77
      .= (|.p1-p.|+|.p2-p.|)^2;
    |.p2-p1.|>r
    proof
      assume |.p2-p1.|<=r;
      then |.p2-p1.|^2<=r^2 by SQUARE_1:15;
      then (r+|.p2-p.|)^2 <= r^2 by A4,Lm2;
      then r^2+2*r*|.p2-p.|+|.p2-p.|^2-r^2<=r^2-r^2 by XREAL_1:9;
      then
A5:   (2*r+|.p2-p.|)*|.p2-p.|<=0;
      |.p2-p.|<>0 by A3,Lm1;
      hence contradiction by A5;
    end;
    then p2 in {p4 : |.p4- |[a,b]| .|>r} by A2;
    then
A6: p2 in outside_of_circle(a,b,r) by JGRAPH_6:def 8;
A7: |.p1 - |[a,b]|.|=0 by A2,Lm1;
    r<>0 by A3,Lm1;
    then p1 in {p4 : |.p4- |[a,b]| .|<r} by A7;
    then p1 in inside_of_circle(a,b,r) by JGRAPH_6:def 6;
    then consider p3 such that
A8: p3 in LSeg(p1,p2) /\ circle(a,b,r) by A6,Lm17;
A9: euc2cpx(p1)<> euc2cpx(p2) by A1,COMPLEX2:79;
A10: euc2cpx(p)<> euc2cpx(p1) & euc2cpx(p)<> euc2cpx(p2) by A3,EUCLID_3:4;
A11: angle(p,p1,p2) = 0
    proof
      assume angle(p,p1,p2) <> 0;
      then
A12:  angle(p,p1,p2) > 0 by COMPLEX2:70;
A13:  angle(p,p1,p2)<2*PI by COMPLEX2:70;
      per cases by A10,A9,COMPLEX2:88;
      suppose
A14:    angle(p,p1,p2)+angle(p1,p2,p)+angle(p2,p,p1) = PI;
A15:    angle(p1,p2,p)>=0 by COMPLEX2:70;
        angle(p,p1,p2)+angle(p1,p2,p)+PI = PI by A1,A14,COMPLEX2:82;
        hence contradiction by A12,A15;
      end;
      suppose
A16:    angle(p,p1,p2)+angle(p1,p2,p)+angle(p2,p,p1) = 5*PI;
        angle(p1,p2,p)<2*PI by COMPLEX2:70;
        then
A17:    angle(p,p1,p2)+angle(p1,p2,p) < 2*PI+2*PI by A13,XREAL_1:8;
        angle(p,p1,p2)+angle(p1,p2,p)+PI = 5*PI by A1,A16,COMPLEX2:82;
        hence contradiction by A17;
      end;
    end;
    p3 in circle(a,b,r) by A8,XBOOLE_0:def 4;
    then p3 in {p4: |.p4- |[a,b]| .|=r} by JGRAPH_6:def 5;
    then
A18: ex p4 st p3=p4 & |.p4 - |[a,b]|.| = r;
    then
A19: |.p3-p1.|=r by EUCLID:53;
    r<>0 by A3,Lm1;
    then
A20: p3<>p1 by A2,A18,Lm1;
A21: p3 in LSeg(p1,p2) by A8,XBOOLE_0:def 4;
    |.p3-p.|^2 = |.p-p1.|^2 + |.p3-p1.|^2 - 2*|.p-p1.|*|.p3-p1.|* cos
    angle(p,p1,p3) by Th7
      .= r^2+r^2 -2*r*r*cos 0 by A21,A19,A20,A11,Th10
      .= 0 by SIN_COS:31;
    then |.p3-p.|=0;
    hence thesis by A21,Lm1;
  end;
end;
