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

for g being Function of (X1 union X2),Y

for x1 being Point of X1

for x2 being Point of X2

for x being Point of (X1 union X2) st x = x1 & x = x2 holds

( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

let X1, X2 be non empty SubSpace of X; :: thesis: for g being Function of (X1 union X2),Y

for x1 being Point of X1

for x2 being Point of X2

for x being Point of (X1 union X2) st x = x1 & x = x2 holds

( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

let g be Function of (X1 union X2),Y; :: thesis: for x1 being Point of X1

for x2 being Point of X2

for x being Point of (X1 union X2) st x = x1 & x = x2 holds

( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

let x1 be Point of X1; :: thesis: for x2 being Point of X2

for x being Point of (X1 union X2) st x = x1 & x = x2 holds

( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

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

( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

let x be Point of (X1 union X2); :: thesis: ( x = x1 & x = x2 implies ( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) ) )

assume that

A1: x = x1 and

A2: x = x2 ; :: thesis: ( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

A3: X2 is SubSpace of X1 union X2 by TSEP_1:22;

A4: X1 is SubSpace of X1 union X2 by TSEP_1:22;

hence ( g is_continuous_at x implies ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) ) by A1, A2, A3, Th74; :: thesis: ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 implies g is_continuous_at x )

thus ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 implies g is_continuous_at x ) :: thesis: verum

for g being Function of (X1 union X2),Y

for x1 being Point of X1

for x2 being Point of X2

for x being Point of (X1 union X2) st x = x1 & x = x2 holds

( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

let X1, X2 be non empty SubSpace of X; :: thesis: for g being Function of (X1 union X2),Y

for x1 being Point of X1

for x2 being Point of X2

for x being Point of (X1 union X2) st x = x1 & x = x2 holds

( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

let g be Function of (X1 union X2),Y; :: thesis: for x1 being Point of X1

for x2 being Point of X2

for x being Point of (X1 union X2) st x = x1 & x = x2 holds

( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

let x1 be Point of X1; :: thesis: for x2 being Point of X2

for x being Point of (X1 union X2) st x = x1 & x = x2 holds

( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

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

( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

let x be Point of (X1 union X2); :: thesis: ( x = x1 & x = x2 implies ( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) ) )

assume that

A1: x = x1 and

A2: x = x2 ; :: thesis: ( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )

A3: X2 is SubSpace of X1 union X2 by TSEP_1:22;

A4: X1 is SubSpace of X1 union X2 by TSEP_1:22;

hence ( g is_continuous_at x implies ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) ) by A1, A2, A3, Th74; :: thesis: ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 implies g is_continuous_at x )

thus ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 implies g is_continuous_at x ) :: thesis: verum

proof

assume that

A5: g | X1 is_continuous_at x1 and

A6: g | X2 is_continuous_at x2 ; :: thesis: g is_continuous_at x

for G being a_neighborhood of g . x ex H being a_neighborhood of x st g .: H c= G

end;A5: g | X1 is_continuous_at x1 and

A6: g | X2 is_continuous_at x2 ; :: thesis: g is_continuous_at x

for G being a_neighborhood of g . x ex H being a_neighborhood of x st g .: H c= G

proof

hence
g is_continuous_at x
; :: thesis: verum
let G be a_neighborhood of g . x; :: thesis: ex H being a_neighborhood of x st g .: H c= G

g . x = (g | X1) . x1 by A1, A4, Th65;

then consider H1 being a_neighborhood of x1 such that

A7: (g | X1) .: H1 c= G by A5;

g . x = (g | X2) . x2 by A2, A3, Th65;

then consider H2 being a_neighborhood of x2 such that

A8: (g | X2) .: H2 c= G by A6;

the carrier of X2 c= the carrier of (X1 union X2) by A3, TSEP_1:4;

then reconsider S2 = H2 as Subset of (X1 union X2) by XBOOLE_1:1;

g .: S2 c= G by A3, A8, Th68;

then A9: S2 c= g " G by FUNCT_2:95;

the carrier of X1 c= the carrier of (X1 union X2) by A4, TSEP_1:4;

then reconsider S1 = H1 as Subset of (X1 union X2) by XBOOLE_1:1;

consider H being a_neighborhood of x such that

A10: H c= H1 \/ H2 by A1, A2, Th16;

take H ; :: thesis: g .: H c= G

g .: S1 c= G by A4, A7, Th68;

then S1 c= g " G by FUNCT_2:95;

then S1 \/ S2 c= g " G by A9, XBOOLE_1:8;

then H c= g " G by A10, XBOOLE_1:1;

hence g .: H c= G by FUNCT_2:95; :: thesis: verum

end;g . x = (g | X1) . x1 by A1, A4, Th65;

then consider H1 being a_neighborhood of x1 such that

A7: (g | X1) .: H1 c= G by A5;

g . x = (g | X2) . x2 by A2, A3, Th65;

then consider H2 being a_neighborhood of x2 such that

A8: (g | X2) .: H2 c= G by A6;

the carrier of X2 c= the carrier of (X1 union X2) by A3, TSEP_1:4;

then reconsider S2 = H2 as Subset of (X1 union X2) by XBOOLE_1:1;

g .: S2 c= G by A3, A8, Th68;

then A9: S2 c= g " G by FUNCT_2:95;

the carrier of X1 c= the carrier of (X1 union X2) by A4, TSEP_1:4;

then reconsider S1 = H1 as Subset of (X1 union X2) by XBOOLE_1:1;

consider H being a_neighborhood of x such that

A10: H c= H1 \/ H2 by A1, A2, Th16;

take H ; :: thesis: g .: H c= G

g .: S1 c= G by A4, A7, Th68;

then S1 c= g " G by FUNCT_2:95;

then S1 \/ S2 c= g " G by A9, XBOOLE_1:8;

then H c= g " G by A10, XBOOLE_1:1;

hence g .: H c= G by FUNCT_2:95; :: thesis: verum