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 V being Subset of (X1 union X2) st
( V is open & x in V & V 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 V being Subset of (X1 union X2) st
( V is open & x in V & V 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 V being Subset of (X1 union X2) st
( V is open & x in V & V 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 V being Subset of (X1 union X2) st
( V is open & x in V & V 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 V being Subset of (X1 union X2) st
( V is open & x in V & V 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 V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
let A1 be a_neighborhood of x1; :: thesis: for A2 being a_neighborhood of x2 ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
let A2 be a_neighborhood of x2; :: thesis: ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= A1 \/ A2 )
consider U1 being Subset of X1 such that
A2: ( U1 is open & U1 c= A1 & x1 in U1 ) by CONNSP_2:8;
consider U2 being Subset of X2 such that
A3: ( U2 is open & U2 c= A2 & x2 in U2 ) by CONNSP_2:8;
consider V being Subset of (X1 union X2) such that
A4: ( V is open & x in V & V c= U1 \/ U2 ) by A1, A2, A3, Th19;
take V ; :: thesis: ( V is open & x in V & V c= A1 \/ A2 )
U1 \/ U2 c= A1 \/ A2 by A2, A3, XBOOLE_1:13;
hence ( V is open & x in V & V c= A1 \/ A2 ) by A4, XBOOLE_1:1; :: thesis: verum