reserve A,B,C,D,E,F,G for Point of TOP-REAL 2;

theorem Th4:
  A,B,C are_mutually_distinct & 0 < angle(A,B,C) < PI implies
  0 < angle(B,C,A) < PI & 0 < angle(C,A,B) < PI
  proof
    assume that
A1: A,B,C are_mutually_distinct and
A2: 0 < angle(A,B,C) < PI;
    set z1 = euc2cpx(A);
    set z2 = euc2cpx(B);
    set z3 = euc2cpx(C);
    z1 <> z2 & z2 <> z3 & z1 <> z3 & 0 < angle(z1,z2,z3) < PI
    by A1,A2,EUCLID_3:4,EUCLID_3:def 4; then
A3: 0 < angle(z2,z3,z1) & 0 < angle(z3,z1,z2) by COMPLEX2:84; then
A4: 0 < angle(B,C,A) & 0 < angle(C,A,B) by EUCLID_3:def 4;
A5: angle(A,B,C) + angle(B,C,A) + angle(C,A,B) = PI by A1,A2,EUCLID_3:47;
    now
      assume PI <= angle(B,C,A) or PI <= angle(C,A,B);
      then per cases;
      suppose PI <= angle(B,C,A);
        then angle(A,B,C) + PI <= angle(A,B,C) + angle(B,C,A) by XREAL_1:6;
        then
A6:     angle(A,B,C) + PI + angle(C,A,B) <= PI by A5,XREAL_1:6;
        0 + PI < angle(A,B,C) + PI by A2,XREAL_1:6;
        hence contradiction by A4,XREAL_1:8,A6;
      end;
      suppose PI <= angle(C,A,B);
        then angle(A,B,C) + PI <= angle(A,B,C) + angle(C,A,B) by XREAL_1:6;
        then
A7:     angle(A,B,C) + PI + angle(B,C,A) <=
        angle(A,B,C) + angle(C,A,B) +angle(B,C,A) by XREAL_1:6;
        0 + PI < angle(A,B,C) + PI by A2,XREAL_1:6;
        hence contradiction by A7,A5,A4,XREAL_1:8;
      end;
    end;
    hence thesis by A3,EUCLID_3:def 4;
  end;
