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

theorem Th14:
  p in LSeg(p1,p2) & p<>p1 & p<>p2 & (angle(p3,p,p1)=PI/2 or angle
  (p3,p,p1)=3/2*PI) implies angle(p1,p,p3)=angle(p3,p,p2)
proof
  assume
A1: p in LSeg(p1,p2) & p<>p1 & p<>p2;
  assume
A2: angle(p3,p,p1)=PI/2 or angle(p3,p,p1)=3/2*PI;
A3: angle(p3,p,p1)=angle(p2,p,p3)
  proof
    per cases by A1,A2,Th13;
    suppose
      angle(p2,p,p3)+angle(p3,p,p1)=PI & angle(p3,p,p1)=PI/2 or angle(
      p2,p,p3)+angle(p3,p,p1)=3*PI & angle(p3,p,p1)=3/2*PI;
      hence thesis;
    end;
    suppose
A4:   angle(p2,p,p3)+angle(p3,p,p1)=PI & angle(p3,p,p1)=3/2*PI;
      (-PI)/2 < 0/2;
      hence thesis by A4,COMPLEX2:70;
    end;
    suppose
A5:   angle(p2,p,p3)+angle(p3,p,p1)=3*PI & angle(p3,p,p1)=PI/2;
      0+2*PI < PI/2+2*PI by XREAL_1:6;
      hence thesis by A5,COMPLEX2:70;
    end;
  end;
  per cases;
  suppose
A6: angle(p3,p,p1)=0;
    then angle(p1,p,p3) = 0 by EUCLID_3:36;
    hence thesis by A3,A6,EUCLID_3:36;
  end;
  suppose
A7: angle(p3,p,p1)<>0;
    then angle(p1,p,p3)=2*PI-angle(p3,p,p1) by EUCLID_3:37;
    hence thesis by A3,A7,EUCLID_3:37;
  end;
end;
