let p1, p2, p3, p4, p5, p6 be Point of (TOP-REAL 2); :: thesis: ( p1,p2,p3 is_a_triangle & p4,p5,p6 is_a_triangle & angle p1,p2,p3 = angle p4,p5,p6 & angle p3,p1,p2 = angle p5,p6,p4 implies ( |.(p2 - p3).| * |.(p4 - p6).| = |.(p3 - p1).| * |.(p5 - p4).| & |.(p2 - p3).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p5 - p4).| & |.(p3 - p1).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p4 - p6).| ) )
assume A1: p1,p2,p3 is_a_triangle ; :: thesis: ( not p4,p5,p6 is_a_triangle or not angle p1,p2,p3 = angle p4,p5,p6 or not angle p3,p1,p2 = angle p5,p6,p4 or ( |.(p2 - p3).| * |.(p4 - p6).| = |.(p3 - p1).| * |.(p5 - p4).| & |.(p2 - p3).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p5 - p4).| & |.(p3 - p1).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p4 - p6).| ) )
then A2: p1,p2,p3 are_mutually_different by Th20;
then A3: p3 <> p2 by ZFMISC_1:def 5;
A4: angle p3,p1,p2 <> PI by A1, Th20;
A5: p3 <> p1 by A2, ZFMISC_1:def 5;
then A6: euc2cpx p3 <> euc2cpx p1 by EUCLID_3:6;
A7: p2 <> p1 by A2, ZFMISC_1:def 5;
then A8: euc2cpx p2 <> euc2cpx p1 by EUCLID_3:6;
assume A9: p4,p5,p6 is_a_triangle ; :: thesis: ( not angle p1,p2,p3 = angle p4,p5,p6 or not angle p3,p1,p2 = angle p5,p6,p4 or ( |.(p2 - p3).| * |.(p4 - p6).| = |.(p3 - p1).| * |.(p5 - p4).| & |.(p2 - p3).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p5 - p4).| & |.(p3 - p1).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p4 - p6).| ) )
then A10: p4,p5,p6 are_mutually_different by Th20;
then A11: p4 <> p5 by ZFMISC_1:def 5;
then A12: euc2cpx p4 <> euc2cpx p5 by EUCLID_3:6;
A13: p5 <> p6 by A10, ZFMISC_1:def 5;
then A14: euc2cpx p5 <> euc2cpx p6 by EUCLID_3:6;
A15: angle p6,p4,p5 <> PI by A9, Th20;
A16: p4 <> p6 by A10, ZFMISC_1:def 5;
then A17: euc2cpx p4 <> euc2cpx p6 by EUCLID_3:6;
assume A18: ( angle p1,p2,p3 = angle p4,p5,p6 & angle p3,p1,p2 = angle p5,p6,p4 ) ; :: thesis: ( |.(p2 - p3).| * |.(p4 - p6).| = |.(p3 - p1).| * |.(p5 - p4).| & |.(p2 - p3).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p5 - p4).| & |.(p3 - p1).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p4 - p6).| )
A19: euc2cpx p3 <> euc2cpx p2 by A3, EUCLID_3:6;
A20: angle p2,p3,p1 = angle p6,p4,p5

proof

per cases ( ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = PI & ((angle p5,p6,p4) + (angle p6,p4,p5)) + (angle p4,p5,p6) = PI ) or ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = 5 * PI & ((angle p5,p6,p4) + (angle p6,p4,p5)) + (angle p4,p5,p6) = 5 * PI ) or ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = PI & ((angle p5,p6,p4) + (angle p6,p4,p5)) + (angle p4,p5,p6) = 5 * PI ) or ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = 5 * PI & ((angle p5,p6,p4) + (angle p6,p4,p5)) + (angle p4,p5,p6) = PI ) ) by A19, A6, A8, A12, A17, A14, COMPLEX2:102;

suppose ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = PI & ((angle p5,p6,p4) + (angle p6,p4,p5)) + (angle p4,p5,p6) = PI ) ; :: thesis: angle p2,p3,p1 = angle p6,p4,p5

hence angle p2,p3,p1 = angle p6,p4,p5 by A18; :: thesis: verum

end;

suppose ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = 5 * PI & ((angle p5,p6,p4) + (angle p6,p4,p5)) + (angle p4,p5,p6) = 5 * PI ) ; :: thesis: angle p2,p3,p1 = angle p6,p4,p5

hence angle p2,p3,p1 = angle p6,p4,p5 by A18; :: thesis: verum

end;

suppose A21: ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = PI & ((angle p5,p6,p4) + (angle p6,p4,p5)) + (angle p4,p5,p6) = 5 * PI ) ; :: thesis: angle p2,p3,p1 = angle p6,p4,p5

angle p6,p4,p5 < 2 * PI by COMPLEX2:84;
then ( angle p2,p3,p1 >= 0 & - (angle p6,p4,p5) > - (2 * PI ) ) by COMPLEX2:84, XREAL_1:26;
then A22: (angle p2,p3,p1) + (- (angle p6,p4,p5)) > 0 + (- (2 * PI )) by XREAL_1:10;
(angle p2,p3,p1) - (angle p6,p4,p5) = - (4 * PI ) by A18, A21;
then 4 * PI < 2 * PI by A22, XREAL_1:26;
then (4 * PI ) / PI < (2 * PI ) / PI by XREAL_1:76;
then 4 < (2 * PI ) / PI by XCMPLX_1:90;
then 4 < 2 by XCMPLX_1:90;
hence angle p2,p3,p1 = angle p6,p4,p5 ; :: thesis: verum

end;

suppose A23: ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = 5 * PI & ((angle p5,p6,p4) + (angle p6,p4,p5)) + (angle p4,p5,p6) = PI ) ; :: thesis: angle p2,p3,p1 = angle p6,p4,p5

( angle p2,p3,p1 < 2 * PI & angle p6,p4,p5 >= 0 ) by COMPLEX2:84;
then (angle p2,p3,p1) + (- (angle p6,p4,p5)) < (2 * PI ) + (- 0 ) by XREAL_1:10;
then (4 * PI ) / PI < (2 * PI ) / PI by A18, A23, XREAL_1:76;
then 4 < (2 * PI ) / PI by XCMPLX_1:90;
then 4 < 2 by XCMPLX_1:90;
hence angle p2,p3,p1 = angle p6,p4,p5 ; :: thesis: verum

end;

end;

end;

( angle p1,p2,p3 <> PI & angle p2,p3,p1 <> PI ) by A1, Th20;
hence |.(p2 - p3).| * |.(p4 - p6).| = |.(p3 - p1).| * |.(p5 - p4).| by A4, A3, A5, A7, A15, A11, A16, A13, A18, Lm19; :: thesis: ( |.(p2 - p3).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p5 - p4).| & |.(p3 - p1).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p4 - p6).| )
A24: ( angle p4,p5,p6 <> PI & angle p5,p6,p4 <> PI ) by A9, Th20;
hence |.(p2 - p3).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p5 - p4).| by A3, A5, A7, A15, A11, A16, A13, A18, A20, Lm19; :: thesis: |.(p3 - p1).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p4 - p6).|
thus |.(p3 - p1).| * |.(p6 - p5).| = |.(p1 - p2).| * |.(p4 - p6).| by A3, A5, A7, A24, A15, A11, A16, A13, A18, A20, Lm19; :: thesis: verum