let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X

for x being Point of (X1 union X2)

for x1 being Point of X1

for x2 being Point of X2 st x1 = x & x2 = x holds

for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let X1, X2 be non empty SubSpace of X; :: thesis: for x being Point of (X1 union X2)

for x1 being Point of X1

for x2 being Point of X2 st x1 = x & x2 = x holds

for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let x be Point of (X1 union X2); :: thesis: for x1 being Point of X1

for x2 being Point of X2 st x1 = x & x2 = x holds

for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let x1 be Point of X1; :: thesis: for x2 being Point of X2 st x1 = x & x2 = x holds

for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let x2 be Point of X2; :: thesis: ( x1 = x & x2 = x implies for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2 )

assume A1: ( x1 = x & x2 = x ) ; :: thesis: for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let A1 be a_neighborhood of x1; :: thesis: for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let A2 be a_neighborhood of x2; :: thesis: ex A being a_neighborhood of x st A c= A1 \/ A2

consider V being Subset of (X1 union X2) such that

A2: ( V is open & x in V ) and

A3: V c= A1 \/ A2 by A1, Th15;

reconsider W = V as a_neighborhood of x by A2, CONNSP_2:3;

take W ; :: thesis: W c= A1 \/ A2

thus W c= A1 \/ A2 by A3; :: thesis: verum

for x being Point of (X1 union X2)

for x1 being Point of X1

for x2 being Point of X2 st x1 = x & x2 = x holds

for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let X1, X2 be non empty SubSpace of X; :: thesis: for x being Point of (X1 union X2)

for x1 being Point of X1

for x2 being Point of X2 st x1 = x & x2 = x holds

for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let x be Point of (X1 union X2); :: thesis: for x1 being Point of X1

for x2 being Point of X2 st x1 = x & x2 = x holds

for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let x1 be Point of X1; :: thesis: for x2 being Point of X2 st x1 = x & x2 = x holds

for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let x2 be Point of X2; :: thesis: ( x1 = x & x2 = x implies for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2 )

assume A1: ( x1 = x & x2 = x ) ; :: thesis: for A1 being a_neighborhood of x1

for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let A1 be a_neighborhood of x1; :: thesis: for A2 being a_neighborhood of x2 ex A being a_neighborhood of x st A c= A1 \/ A2

let A2 be a_neighborhood of x2; :: thesis: ex A being a_neighborhood of x st A c= A1 \/ A2

consider V being Subset of (X1 union X2) such that

A2: ( V is open & x in V ) and

A3: V c= A1 \/ A2 by A1, Th15;

reconsider W = V as a_neighborhood of x by A2, CONNSP_2:3;

take W ; :: thesis: W c= A1 \/ A2

thus W c= A1 \/ A2 by A3; :: thesis: verum