let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace of V

for x being object holds

( x in W1 + W2 iff ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 ) )

let W1, W2 be Subspace of V; :: thesis: for x being object holds

( x in W1 + W2 iff ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 ) )

let x be object ; :: thesis: ( x in W1 + W2 iff ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 ) )

thus ( x in W1 + W2 implies ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 ) ) :: thesis: ( ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 ) implies x in W1 + W2 )

x in { (v + u) where u, v is VECTOR of V : ( v in W1 & u in W2 ) } by A2;

then x in the carrier of (W1 + W2) by Def1;

hence x in W1 + W2 by STRUCT_0:def 5; :: thesis: verum

for x being object holds

( x in W1 + W2 iff ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 ) )

let W1, W2 be Subspace of V; :: thesis: for x being object holds

( x in W1 + W2 iff ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 ) )

let x be object ; :: thesis: ( x in W1 + W2 iff ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 ) )

thus ( x in W1 + W2 implies ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 ) ) :: thesis: ( ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 ) implies x in W1 + W2 )

proof

given v1, v2 being VECTOR of V such that A2:
( v1 in W1 & v2 in W2 & x = v1 + v2 )
; :: thesis: x in W1 + W2
assume
x in W1 + W2
; :: thesis: ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 )

then x in the carrier of (W1 + W2) by STRUCT_0:def 5;

then x in { (v + u) where u, v is VECTOR of V : ( v in W1 & u in W2 ) } by Def1;

then consider v2, v1 being VECTOR of V such that

A1: ( x = v1 + v2 & v1 in W1 & v2 in W2 ) ;

take v1 ; :: thesis: ex v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 )

take v2 ; :: thesis: ( v1 in W1 & v2 in W2 & x = v1 + v2 )

thus ( v1 in W1 & v2 in W2 & x = v1 + v2 ) by A1; :: thesis: verum

end;( v1 in W1 & v2 in W2 & x = v1 + v2 )

then x in the carrier of (W1 + W2) by STRUCT_0:def 5;

then x in { (v + u) where u, v is VECTOR of V : ( v in W1 & u in W2 ) } by Def1;

then consider v2, v1 being VECTOR of V such that

A1: ( x = v1 + v2 & v1 in W1 & v2 in W2 ) ;

take v1 ; :: thesis: ex v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & x = v1 + v2 )

take v2 ; :: thesis: ( v1 in W1 & v2 in W2 & x = v1 + v2 )

thus ( v1 in W1 & v2 in W2 & x = v1 + v2 ) by A1; :: thesis: verum

x in { (v + u) where u, v is VECTOR of V : ( v in W1 & u in W2 ) } by A2;

then x in the carrier of (W1 + W2) by Def1;

hence x in W1 + W2 by STRUCT_0:def 5; :: thesis: verum