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

theorem Th33:
  p1 in circle(a,b,r) & p2 in circle(a,b,r) & p in circle(a,b,r) &
pc = |[a,b]| & p1<>p & p2<>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
A4: pc = |[a,b]|;
  assume that
A5: p1<>p and
A6: p2<>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;
A11: |.p2-pc.|=r by A2,A4,TOPREAL9:43;
    |.p1 - pc.| >= 0;
    then r > 0 by A1,A4,A10,TOPREAL9:43;
    then consider p3 such that
    p<>p3 and
A12: p3 in circle(a,b,r) and
A13: |[a,b]| in LSeg(p,p3) by A3,Th32;
    per cases;
    suppose
      p2=p3;
      hence thesis by A1,A3,A4,A5,A12,A13,Th31;
    end;
    suppose
A14:  p2<>p3;
A15:  angle(p2,pc,p3)<>0
      proof
        set z3=euc2cpx(p3-pc);
        set z2=euc2cpx(p2-pc);
        assume angle(p2,pc,p3)=0;
        then
A16:    Arg(p2-pc)=Arg(p3-pc) by EUCLID_3:36;
A17:    |.p2-pc.|=|.p3-pc.| by A4,A11,A12,TOPREAL9:43;
A18:    |.z2.| = |.p2-pc.| by EUCLID_3:25
          .= |.z3.| by A17,EUCLID_3:25;
A19:    z2 = |.z2.|*cos Arg z2+|.z2.|*sin Arg z2 *<i> by COMPTRIG:62
          .= z3 by A16,A18,COMPTRIG:62;
        p2 = p2 + 0.TOP-REAL 2 by RLVECT_1:4
          .= p2 +(pc+(-pc)) by RLVECT_1:5
          .= p2 +(-pc) + pc by RLVECT_1:def 3
          .= p3-pc + pc by A19,EUCLID_3:4
          .= p3 +(pc+(-pc)) by RLVECT_1:def 3
          .= p3 + 0.TOP-REAL 2 by RLVECT_1:5
          .= p3 by RLVECT_1:4;
        hence contradiction by A14;
      end;
      2*angle(p2,p,p3) = angle(p2,pc,p3) or 2*(angle(p2,p,p3) - PI) =
      angle(p2,pc,p3) by A2,A3,A4,A6,A12,A13,Th31;
      then
A20:  angle(p2,p,p3)<>0 by A15,COMPLEX2:70;
A21:  2*(angle(p2,p,p3) - PI) = angle(p2,pc,p3) implies 2*angle(p3,p,p2)
      = angle(p3,pc,p2)
      proof
        assume
A22:    2*(angle(p2,p,p3) - PI) = angle(p2,pc,p3);
        thus 2*angle(p3,p,p2) = 2*(2*PI-angle(p2,p,p3)) by A20,EUCLID_3:37
          .= 2*PI -2*(angle(p2,p,p3)-PI)
          .= angle(p3,pc,p2) by A15,A22,EUCLID_3:37;
      end;
A23:  angle(p3,p,p2) = 2*PI-angle(p2,p,p3) by A20,EUCLID_3:37;
A24:  2*angle(p2,p,p3) = angle(p2,pc,p3) implies 2*(angle(p3,p,p2) -PI) =
      angle(p3,pc,p2)
      proof
        assume 2*angle(p2,p,p3) = angle(p2,pc,p3);
        hence 2*(angle(p3,p,p2)-PI) = 2*PI - angle(p2,pc,p3) by A23
          .= angle(p3,pc,p2) by A15,EUCLID_3:37;
      end;
A25:  angle(p1,p,p2) = angle(p1,p,p3)+angle(p3,p,p2) or angle(p1,p,p2)+2*
      PI = angle(p1,p,p3)+angle(p3,p,p2) by Th4;
      per cases by Th4;
      suppose
A26:    angle(p1,pc,p2) = angle(p1,pc,p3)+angle(p3,pc,p2);
        per cases by A1,A2,A3,A4,A5,A6,A12,A13,A24,A21,Th31;
        suppose
          2*angle(p1,p,p3) = angle(p1,pc,p3) & 2*(angle(p3,p,p2)-PI)
          = angle(p3,pc,p2);
          then angle(p1,pc,p2) = 2*angle(p1,p,p2) - 2*PI or angle(p1,pc,p2) =
          2*angle(p1,p,p2)+2*PI by A25,A26;
          hence thesis by Lm3;
        end;
        suppose
          2*angle(p1,p,p3) = angle(p1,pc,p3) & 2*angle(p3,p,p2) =
          angle(p3,pc,p2);
          then
          angle(p1,pc,p2) = 2*angle(p1,p,p2) or angle(p1,pc,p2) = 2*angle
          (p1,p,p2)+4*PI by A25,A26;
          hence thesis by Lm4;
        end;
        suppose
          2*(angle(p1,p,p3) - PI) = angle(p1,pc,p3) & 2*(angle(p3,p,
          p2)-PI) = angle(p3,pc,p2);
          then
          angle(p1,pc,p2) = 2*(angle(p1,p,p3)+angle(p3,p,p2))-4*PI by A26;
          hence thesis by A25,Lm5;
        end;
        suppose
          2*(angle(p1,p,p3) - PI) = angle(p1,pc,p3) & 2*angle(p3,p,p2
          ) = angle(p3,pc,p2);
          then
          angle(p1,pc,p2) = 2*angle(p1,p,p2)-2*PI or angle(p1,pc,p2) = 2*
          angle(p1,p,p2)+2*PI by A25,A26;
          hence thesis by Lm3;
        end;
      end;
      suppose
A27:    angle(p1,pc,p2)+2*PI = angle(p1,pc,p3)+angle(p3,pc,p2);
        per cases by A1,A2,A3,A4,A5,A6,A12,A13,A24,A21,Th31;
        suppose
          2*angle(p1,p,p3) = angle(p1,pc,p3) & 2*(angle(p3,p,p2)-PI)
          = angle(p3,pc,p2);
          then
          angle(p1,pc,p2) = 2*(angle(p1,p,p3)+angle(p3,p,p2))-4*PI by A27;
          hence thesis by A25,Lm5;
        end;
        suppose
          2*angle(p1,p,p3) = angle(p1,pc,p3) & 2*angle(p3,p,p2) =
          angle(p3,pc,p2);
          then
          angle(p1,pc,p2) = 2*angle(p1,p,p2)-2*PI or angle(p1,pc,p2) = 2*
          angle(p1,p,p2)+2*PI by A25,A27;
          hence thesis by Lm3;
        end;
        suppose
          2*(angle(p1,p,p3) - PI) = angle(p1,pc,p3) & 2*(angle(p3,p,
          p2)-PI) = angle(p3,pc,p2);
          then
          angle(p1,pc,p2) = 2*(angle(p1,p,p3)+angle(p3,p,p2))-6*PI by A27;
          hence thesis by A25,Lm6;
        end;
        suppose
          2*(angle(p1,p,p3) - PI) = angle(p1,pc,p3) & 2*angle(p3,p,p2
          ) = angle(p3,pc,p2);
          then
          angle(p1,pc,p2) = 2*(angle(p1,p,p3)+angle(p3,p,p2))-4*PI by A27;
          hence thesis by A25,Lm5;
        end;
      end;
    end;
  end;
end;
