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

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 A1, Th70;

hence f | (X1 union X2) is_continuous_at x by A2, A3, Th111; :: thesis: verum

end;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 A1, Th70;

hence f | (X1 union X2) is_continuous_at x by A2, A3, Th111; :: thesis: verum