let p, q be Point of (TOP-REAL 2); :: thesis: for R being Subset of (TOP-REAL 2) st R is being_Region & p in R & q in R & p <> q holds
ex P being Subset of (TOP-REAL 2) st
( P is_S-P_arc_joining p,q & P c= R )
let R be Subset of (TOP-REAL 2); :: thesis: ( R is being_Region & p in R & q in R & p <> q implies ex P being Subset of (TOP-REAL 2) st
( P is_S-P_arc_joining p,q & P c= R ) )
set RR = { q2 where q2 is Point of (TOP-REAL 2) : ( q2 = p or ex P1 being Subset of (TOP-REAL 2) st
( P1 is_S-P_arc_joining p,q2 & P1 c= R ) ) } ;
{ q2 where q2 is Point of (TOP-REAL 2) : ( q2 = p or ex P1 being Subset of (TOP-REAL 2) st
( P1 is_S-P_arc_joining p,q2 & P1 c= R ) ) } c= the carrier of (TOP-REAL 2)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { q2 where q2 is Point of (TOP-REAL 2) : ( q2 = p or ex P1 being Subset of (TOP-REAL 2) st
( P1 is_S-P_arc_joining p,q2 & P1 c= R ) ) } or x in the carrier of (TOP-REAL 2) )
assume x in { q2 where q2 is Point of (TOP-REAL 2) : ( q2 = p or ex P1 being Subset of (TOP-REAL 2) st
( P1 is_S-P_arc_joining p,q2 & P1 c= R ) ) } ; :: thesis: x in the carrier of (TOP-REAL 2)
then ex q2 being Point of (TOP-REAL 2) st
( q2 = x & ( q2 = p or ex P1 being Subset of (TOP-REAL 2) st
( P1 is_S-P_arc_joining p,q2 & P1 c= R ) ) ) ;
hence x in the carrier of (TOP-REAL 2) ; :: thesis: verum

end;

then reconsider RR = { q2 where q2 is Point of (TOP-REAL 2) : ( q2 = p or ex P1 being Subset of (TOP-REAL 2) st
( P1 is_S-P_arc_joining p,q2 & P1 c= R ) ) } as Subset of (TOP-REAL 2) ;
assume that
A1: ( R is being_Region & p in R ) and
A2: q in R and
A3: p <> q ; :: thesis: ex P being Subset of (TOP-REAL 2) st
( P is_S-P_arc_joining p,q & P c= R )
R c= RR by A1, Th27;
then q in RR by A2;
then ex q1 being Point of (TOP-REAL 2) st
( q1 = q & ( q1 = p or ex P1 being Subset of (TOP-REAL 2) st
( P1 is_S-P_arc_joining p,q1 & P1 c= R ) ) ) ;
hence ex P being Subset of (TOP-REAL 2) st
( P is_S-P_arc_joining p,q & P c= R ) by A3; :: thesis: verum