let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2

let W1, W2 be Subspace of V; :: thesis: the carrier of ((W1 /\ W2) + W2) = the carrier of W2

thus the carrier of ((W1 /\ W2) + W2) c= the carrier of W2 :: according to XBOOLE_0:def 10 :: thesis: the carrier of W2 c= the carrier of ((W1 /\ W2) + W2)

the carrier of W2 c= the carrier of (W2 + (W1 /\ W2)) by Lm2;

then A4: the carrier of W2 c= the carrier of ((W1 /\ W2) + W2) by Lm1;

assume x in the carrier of W2 ; :: thesis: x in the carrier of ((W1 /\ W2) + W2)

hence x in the carrier of ((W1 /\ W2) + W2) by A4; :: thesis: verum

let W1, W2 be Subspace of V; :: thesis: the carrier of ((W1 /\ W2) + W2) = the carrier of W2

thus the carrier of ((W1 /\ W2) + W2) c= the carrier of W2 :: according to XBOOLE_0:def 10 :: thesis: the carrier of W2 c= the carrier of ((W1 /\ W2) + W2)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of W2 or x in the carrier of ((W1 /\ W2) + W2) )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of ((W1 /\ W2) + W2) or x in the carrier of W2 )

assume x in the carrier of ((W1 /\ W2) + W2) ; :: thesis: x in the carrier of W2

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

then consider v, u being VECTOR of V such that

A1: x = u + v and

A2: u in W1 /\ W2 and

A3: v in W2 ;

u in W2 by A2, Th3;

then u + v in W2 by A3, RLSUB_1:20;

hence x in the carrier of W2 by A1, STRUCT_0:def 5; :: thesis: verum

end;assume x in the carrier of ((W1 /\ W2) + W2) ; :: thesis: x in the carrier of W2

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

then consider v, u being VECTOR of V such that

A1: x = u + v and

A2: u in W1 /\ W2 and

A3: v in W2 ;

u in W2 by A2, Th3;

then u + v in W2 by A3, RLSUB_1:20;

hence x in the carrier of W2 by A1, STRUCT_0:def 5; :: thesis: verum

the carrier of W2 c= the carrier of (W2 + (W1 /\ W2)) by Lm2;

then A4: the carrier of W2 c= the carrier of ((W1 /\ W2) + W2) by Lm1;

assume x in the carrier of W2 ; :: thesis: x in the carrier of ((W1 /\ W2) + W2)

hence x in the carrier of ((W1 /\ W2) + W2) by A4; :: thesis: verum