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

theorem Th31:
  p1 in circle(a,b,r) & p2 in circle(a,b,r) & p in circle(a,b,r) &
pc = |[a,b]| & pc in LSeg(p,p2) & p1<>p implies 2*angle(p1,p,p2) = angle(p1,pc,
  p2) or 2*(angle(p1,p,p2) - PI) = angle(p1,pc,p2)
proof
  assume
A1: p1 in circle(a,b,r);
  assume
A2: p2 in circle(a,b,r);
  assume
A3: p in circle(a,b,r);
  assume that
A4: pc = |[a,b]| and
A5: pc in LSeg(p,p2);
  assume
A6: p1<>p;
  per cases;
  suppose
A7: r=0;
    then |.p1-pc.|=0 by A1,A4,TOPREAL9:43;
    then
A8: p1=pc by Lm1;
A9: |.p2-pc.|=0 by A2,A4,A7,TOPREAL9:43;
    then p2=pc by Lm1;
    then 2*angle(p1,p,p2) = 2*0 by A8,COMPLEX2:79
      .= angle(pc,pc,pc) by COMPLEX2:79;
    hence thesis by A9,A8,Lm1;
  end;
  suppose
A10: r<>0;
    |.p2-pc.|=r by A2,A4,TOPREAL9:43;
    then
A11: pc<>p2 by A10,Lm1;
A12: euc2cpx(p1)<> euc2cpx(p) by A6,EUCLID_3:4;
A13: |.p1-pc.|=r by A1,A4,TOPREAL9:43;
    then pc<>p1 by A10,Lm1;
    then
A14: euc2cpx(pc)<> euc2cpx(p1) by EUCLID_3:4;
A15: |.p-pc.|=r by A3,A4,TOPREAL9:43;
    then
A16: pc<>p by A10,Lm1;
    then
A17: angle(p1,p,p2)=angle(p1,p,pc) by A5,Th10;
    |.pc-p1.|=|.p-pc.| by A13,A15,Lm2;
    then
A18: angle(pc,p1,p)=angle(p1,p,pc) by A6,Th16;
    euc2cpx(pc)<> euc2cpx(p) by A16,EUCLID_3:4;
    then
A19: angle(pc,p1,p)+angle(p1,p,pc)+angle(p,pc,p1)=PI or angle(pc,p1,p)+
    angle(p1,p,pc)+angle(p,pc,p1)=5*PI by A14,A12,COMPLEX2:88;
    per cases by A5,A16,A11,A19,A18,A17,Th13;
    suppose
      angle(p,pc,p1)+angle(p1,pc,p2)=PI & 2*angle(p1,p,p2)+angle(p,pc ,p1)=PI;
      hence thesis;
    end;
    suppose
A20:  angle(p,pc,p1)+angle(p1,pc,p2)=3*PI & 2*angle(p1,p,p2)+angle(p,
      pc,p1)=PI;
      angle(p1,pc,p2)<2*PI & angle(p1,p,p2)>=0 by COMPLEX2:70;
      then -2*angle(p1,p,p2)+angle(p1,pc,p2)<0+2*PI by XREAL_1:8;
      hence thesis by A20;
    end;
    suppose
A21:  angle(p,pc,p1)+angle(p1,pc,p2)=PI & 2*angle(p1,p,p2)+angle(p,pc
      ,p1)=5*PI;
      angle(p1,pc,p2)>=0 & angle(p1,p,p2)*2 < (2*PI)*2 by COMPLEX2:70
,XREAL_1:68;
      then 2*angle(p1,p,p2)+(-angle(p1,pc,p2))<(2*PI)*2+0 by XREAL_1:8;
      hence thesis by A21;
    end;
    suppose
      angle(p,pc,p1)+angle(p1,pc,p2)=3*PI & 2*angle(p1,p,p2)+angle(p,
      pc,p1)=5*PI;
      hence thesis;
    end;
  end;
end;
