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

proof

per cases ( ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = PI & ((angle p6,p4,p5) + (angle p4,p5,p6)) + (angle p5,p6,p4) = PI ) or ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = 5 * PI & ((angle p6,p4,p5) + (angle p4,p5,p6)) + (angle p5,p6,p4) = 5 * PI ) or ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = PI & ((angle p6,p4,p5) + (angle p4,p5,p6)) + (angle p5,p6,p4) = 5 * PI ) or ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = 5 * PI & ((angle p6,p4,p5) + (angle p4,p5,p6)) + (angle p5,p6,p4) = PI ) ) by A17, A6, A4, A11, A15, A13, COMPLEX2:102;

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

hence angle p2,p3,p1 = angle p5,p6,p4 by A16; :: thesis: verum

end;

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

hence angle p2,p3,p1 = angle p5,p6,p4 by A16; :: thesis: verum

end;

suppose A19: ( ((angle p3,p1,p2) + (angle p1,p2,p3)) + (angle p2,p3,p1) = PI & ((angle p6,p4,p5) + (angle p4,p5,p6)) + (angle p5,p6,p4) = 5 * PI ) ; :: thesis: angle p2,p3,p1 = angle p5,p6,p4

angle p5,p6,p4 < 2 * PI by COMPLEX2:84;
then ( angle p2,p3,p1 >= 0 & - (angle p5,p6,p4) > - (2 * PI ) ) by COMPLEX2:84, XREAL_1:26;
then A20: (angle p2,p3,p1) + (- (angle p5,p6,p4)) > 0 + (- (2 * PI )) by XREAL_1:10;
(angle p2,p3,p1) - (angle p5,p6,p4) = - (4 * PI ) by A16, A19;
then 4 * PI < 2 * PI by A20, 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 p5,p6,p4 ; :: thesis: verum

end;

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

( angle p2,p3,p1 < 2 * PI & angle p5,p6,p4 >= 0 ) by COMPLEX2:84;
then (angle p2,p3,p1) + (- (angle p5,p6,p4)) < (2 * PI ) + (- 0 ) by XREAL_1:10;
then (4 * PI ) / PI < (2 * PI ) / PI by A16, A21, 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 p5,p6,p4 ; :: thesis: verum

end;

end;

end;

A22: angle p6,p4,p5 <> PI by A7, Th20;
hence |.(p3 - p2).| * |.(p4 - p6).| = |.(p1 - p3).| * |.(p6 - p5).| by A2, A5, A3, A8, A10, A14, A12, A16, A18, Lm18; :: thesis: ( |.(p3 - p2).| * |.(p5 - p4).| = |.(p2 - p1).| * |.(p6 - p5).| & |.(p1 - p3).| * |.(p5 - p4).| = |.(p2 - p1).| * |.(p4 - p6).| )
thus |.(p3 - p2).| * |.(p5 - p4).| = |.(p2 - p1).| * |.(p6 - p5).| by A2, A5, A3, A8, A22, A10, A14, A12, A16, A18, Lm18; :: thesis: |.(p1 - p3).| * |.(p5 - p4).| = |.(p2 - p1).| * |.(p4 - p6).|
thus |.(p1 - p3).| * |.(p5 - p4).| = |.(p2 - p1).| * |.(p4 - p6).| by A2, A5, A3, A8, A22, A10, A14, A12, A16, A18, Lm18; :: thesis: verum