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

theorem Th4:
  angle(p1,p4,p2)+angle(p2,p4,p3)=angle(p1,p4,p3) or angle(p1,p4,p2
  )+angle(p2,p4,p3)=angle(p1,p4,p3) + 2*PI
proof
  set c1 = euc2cpx(p1-p4);
  set c2 = euc2cpx(p2-p4);
  set c3 = euc2cpx(p3-p4);
A1: angle(c1,c2) + angle(c2,c3) = angle(c1,c3) or angle(c1,c2) + angle(c2,c3
  ) = angle(c1,c3) + 2*PI by Lm14;
  angle(p1,p4,p2)+angle(p2,p4,p3)=angle(c1,c2)+angle(p2,p4,p3) by Lm7
    .= angle(c1,c2)+angle(c2,c3) by Lm7;
  hence thesis by A1,Lm7;
end;
