let P, Q be Subset of (); :: thesis: for p1, p2 being Point of () st p1 in Q & P /\ Q is closed & P is_an_arc_of p1,p2 holds
First_Point (P,p1,p2,Q) = p1

let p1, p2 be Point of (); :: thesis: ( p1 in Q & P /\ Q is closed & P is_an_arc_of p1,p2 implies First_Point (P,p1,p2,Q) = p1 )
assume that
A1: p1 in Q and
A2: P /\ Q is closed and
A3: P is_an_arc_of p1,p2 ; :: thesis: First_Point (P,p1,p2,Q) = p1
A4: for g being Function of I,(() | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = p1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds
not g . t in Q
proof
let g be Function of I,(() | P); :: thesis: for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = p1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds
not g . t in Q

let s2 be Real; :: thesis: ( g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = p1 & 0 <= s2 & s2 <= 1 implies for t being Real st 0 <= t & t < s2 holds
not g . t in Q )

assume that
A5: g is being_homeomorphism and
A6: g . 0 = p1 and
g . 1 = p2 and
A7: g . s2 = p1 and
A8: ( 0 <= s2 & s2 <= 1 ) ; :: thesis: for t being Real st 0 <= t & t < s2 holds
not g . t in Q

A9: g is one-to-one by ;
let t be Real; :: thesis: ( 0 <= t & t < s2 implies not g . t in Q )
assume A10: ( 0 <= t & t < s2 ) ; :: thesis: not g . t in Q
A11: dom g = [#] I by
.= the carrier of I ;
then A12: 0 in dom g by BORSUK_1:43;
s2 in dom g by ;
hence not g . t in Q by ; :: thesis: verum
end;
p1 in P by ;
then ( p1 in P /\ Q & P meets Q ) by ;
hence First_Point (P,p1,p2,Q) = p1 by A2, A3, A4, Def1; :: thesis: verum