let p, q be Point of (); :: thesis: for R being Subset of () st R is being_Region & p in R & q in R & p <> q holds
ex P being Subset of () st
( P is_S-P_arc_joining p,q & P c= R )

let R be Subset of (); :: thesis: ( R is being_Region & p in R & q in R & p <> q implies ex P being Subset of () st
( P is_S-P_arc_joining p,q & P c= R ) )

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

assume x in { q2 where q2 is Point of () : ( q2 = p or ex P1 being Subset of () st
( P1 is_S-P_arc_joining p,q2 & P1 c= R ) )
}
; :: thesis: x in the carrier of ()
then ex q2 being Point of () st
( q2 = x & ( q2 = p or ex P1 being Subset of () st
( P1 is_S-P_arc_joining p,q2 & P1 c= R ) ) ) ;
hence x in the carrier of () ; :: thesis: verum
end;
then reconsider RR = { q2 where q2 is Point of () : ( q2 = p or ex P1 being Subset of () st
( P1 is_S-P_arc_joining p,q2 & P1 c= R ) )
}
as Subset of () ;
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 () st
( P is_S-P_arc_joining p,q & P c= R )

R c= RR by ;
then q in RR by A2;
then ex q1 being Point of () st
( q1 = q & ( q1 = p or ex P1 being Subset of () st
( P1 is_S-P_arc_joining p,q1 & P1 c= R ) ) ) ;
hence ex P being Subset of () st
( P is_S-P_arc_joining p,q & P c= R ) by A3; :: thesis: verum