let X, Y be non empty TopSpace; :: thesis: for f being Function of X,Y
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 x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let f be Function of X,Y; :: 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 x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

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 x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

A1: ( X1 is SubSpace of X1 union X2 & X2 is SubSpace of X1 union X2 ) by TSEP_1:22;
let x be Point of (X1 union X2); :: thesis: for x1 being Point of X1
for x2 being Point of X2 st x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x1 be Point of X1; :: thesis: for x2 being Point of X2 st x = x1 & x = x2 holds
( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x2 be Point of X2; :: thesis: ( x = x1 & x = x2 implies ( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) )
assume A2: ( x = x1 & x = x2 ) ; :: thesis: ( f | (X1 union X2) is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )
thus ( f | (X1 union X2) is_continuous_at x implies ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) by A2, A1, Th75; :: thesis: ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f | (X1 union X2) is_continuous_at x )
thus ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f | (X1 union X2) is_continuous_at x ) :: thesis: verum
proof
set g = f | (X1 union X2);
assume A3: ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ; :: thesis: f | (X1 union X2) is_continuous_at x
( (f | (X1 union X2)) | X1 = f | X1 & (f | (X1 union X2)) | X2 = f | X2 ) by ;
hence f | (X1 union X2) is_continuous_at x by ; :: thesis: verum
end;