let IT1, IT2 be Point of (TOP-REAL 2); :: thesis: ( IT1 in P /\ Q & ( for g being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = IT1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 1 >= t & t > s2 holds
not g . t in Q ) & IT2 in P /\ Q & ( for g being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = IT2 & 0 <= s2 & s2 <= 1 holds
for t being Real st 1 >= t & t > s2 holds
not g . t in Q ) implies IT1 = IT2 )
A5: P /\ Q c= P by XBOOLE_1:17;
assume that
A6: IT1 in P /\ Q and
A7: for g1 being Function of I[01],((TOP-REAL 2) | P)
for s1 being Real st g1 is being_homeomorphism & g1 . 0 = p1 & g1 . 1 = p2 & g1 . s1 = IT1 & 0 <= s1 & s1 <= 1 holds
for t being Real st 1 >= t & t > s1 holds
not g1 . t in Q ; :: thesis: ( not IT2 in P /\ Q or ex g being Function of I[01],((TOP-REAL 2) | P) ex s2 being Real st
( g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = IT2 & 0 <= s2 & s2 <= 1 & ex t being Real st
( 1 >= t & t > s2 & g . t in Q ) ) or IT1 = IT2 )
assume that
A8: IT2 in P /\ Q and
A9: for g2 being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g2 is being_homeomorphism & g2 . 0 = p1 & g2 . 1 = p2 & g2 . s2 = IT2 & 0 <= s2 & s2 <= 1 holds
for t being Real st 1 >= t & t > s2 holds
not g2 . t in Q ; :: thesis: IT1 = IT2
consider g being Function of I[01],((TOP-REAL 2) | P) such that
A10: g is being_homeomorphism and
A11: ( g . 0 = p1 & g . 1 = p2 ) by A2, TOPREAL1:def 1;
A12: rng g = [#] ((TOP-REAL 2) | P) by A10, TOPS_2:def 5
.= P by PRE_TOPC:def 5 ;
then consider ss1 being object such that
A13: ss1 in dom g and
A14: g . ss1 = IT1 by A5, A6, FUNCT_1:def 3;
reconsider ss1 = ss1 as Real by A13, BORSUK_1:40;
A15: 0 <= ss1 by A13, BORSUK_1:43;
A16: ( P /\ Q c= Q & ss1 <= 1 ) by A13, BORSUK_1:43, XBOOLE_1:17;
consider ss2 being object such that
A17: ss2 in dom g and
A18: g . ss2 = IT2 by A5, A8, A12, FUNCT_1:def 3;
reconsider ss2 = ss2 as Real by A17, BORSUK_1:40;
A19: 0 <= ss2 by A17, BORSUK_1:43;
A20: ss2 <= 1 by A17, BORSUK_1:43;

per cases ( ss1 < ss2 or ss1 = ss2 or ss1 > ss2 ) by XXREAL_0:1;

suppose ss1 < ss2 ; :: thesis: IT1 = IT2

hence IT1 = IT2 by A7, A8, A10, A11, A14, A18, A15, A16, A20; :: thesis: verum

end;

suppose ss1 = ss2 ; :: thesis: IT1 = IT2

hence IT1 = IT2 by A14, A18; :: thesis: verum

end;

suppose ss1 > ss2 ; :: thesis: IT1 = IT2

hence IT1 = IT2 by A6, A9, A10, A11, A14, A18, A16, A19, A20; :: thesis: verum

end;

end;