let n be Nat; :: thesis: for P1, P being Subset of (TOP-REAL n)

for Q being Subset of ((TOP-REAL n) | P)

for p1, p2 being Point of (TOP-REAL n) st P1 is_an_arc_of p1,p2 & P1 c= P & Q = P1 \ {p1,p2} holds

Q is connected

let P1, P be Subset of (TOP-REAL n); :: thesis: for Q being Subset of ((TOP-REAL n) | P)

for p1, p2 being Point of (TOP-REAL n) st P1 is_an_arc_of p1,p2 & P1 c= P & Q = P1 \ {p1,p2} holds

Q is connected

let Q be Subset of ((TOP-REAL n) | P); :: thesis: for p1, p2 being Point of (TOP-REAL n) st P1 is_an_arc_of p1,p2 & P1 c= P & Q = P1 \ {p1,p2} holds

Q is connected

let p1, p2 be Point of (TOP-REAL n); :: thesis: ( P1 is_an_arc_of p1,p2 & P1 c= P & Q = P1 \ {p1,p2} implies Q is connected )

assume that

A1: P1 is_an_arc_of p1,p2 and

A2: P1 c= P and

A3: Q = P1 \ {p1,p2} ; :: thesis: Q is connected

[#] ((TOP-REAL n) | P1) = P1 by PRE_TOPC:def 5;

then reconsider Q9 = Q as Subset of ((TOP-REAL n) | P1) by A3, XBOOLE_1:36;

Q9 is connected by A1, A3, Th40;

hence Q is connected by A2, Th41; :: thesis: verum

for Q being Subset of ((TOP-REAL n) | P)

for p1, p2 being Point of (TOP-REAL n) st P1 is_an_arc_of p1,p2 & P1 c= P & Q = P1 \ {p1,p2} holds

Q is connected

let P1, P be Subset of (TOP-REAL n); :: thesis: for Q being Subset of ((TOP-REAL n) | P)

for p1, p2 being Point of (TOP-REAL n) st P1 is_an_arc_of p1,p2 & P1 c= P & Q = P1 \ {p1,p2} holds

Q is connected

let Q be Subset of ((TOP-REAL n) | P); :: thesis: for p1, p2 being Point of (TOP-REAL n) st P1 is_an_arc_of p1,p2 & P1 c= P & Q = P1 \ {p1,p2} holds

Q is connected

let p1, p2 be Point of (TOP-REAL n); :: thesis: ( P1 is_an_arc_of p1,p2 & P1 c= P & Q = P1 \ {p1,p2} implies Q is connected )

assume that

A1: P1 is_an_arc_of p1,p2 and

A2: P1 c= P and

A3: Q = P1 \ {p1,p2} ; :: thesis: Q is connected

[#] ((TOP-REAL n) | P1) = P1 by PRE_TOPC:def 5;

then reconsider Q9 = Q as Subset of ((TOP-REAL n) | P1) by A3, XBOOLE_1:36;

Q9 is connected by A1, A3, Th40;

hence Q is connected by A2, Th41; :: thesis: verum