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

for W1 being strict Subspace of V holds

( W1 is Subspace of W2 iff W1 /\ W2 = W1 )

let W2 be Subspace of V; :: thesis: for W1 being strict Subspace of V holds

( W1 is Subspace of W2 iff W1 /\ W2 = W1 )

let W1 be strict Subspace of V; :: thesis: ( W1 is Subspace of W2 iff W1 /\ W2 = W1 )

thus ( W1 is Subspace of W2 implies W1 /\ W2 = W1 ) :: thesis: ( W1 /\ W2 = W1 implies W1 is Subspace of W2 )

for W1 being strict Subspace of V holds

( W1 is Subspace of W2 iff W1 /\ W2 = W1 )

let W2 be Subspace of V; :: thesis: for W1 being strict Subspace of V holds

( W1 is Subspace of W2 iff W1 /\ W2 = W1 )

let W1 be strict Subspace of V; :: thesis: ( W1 is Subspace of W2 iff W1 /\ W2 = W1 )

thus ( W1 is Subspace of W2 implies W1 /\ W2 = W1 ) :: thesis: ( W1 /\ W2 = W1 implies W1 is Subspace of W2 )

proof

thus
( W1 /\ W2 = W1 implies W1 is Subspace of W2 )
by Th16; :: thesis: verum
assume
W1 is Subspace of W2
; :: thesis: W1 /\ W2 = W1

then A1: the carrier of W1 c= the carrier of W2 by RLSUB_1:def 2;

the carrier of (W1 /\ W2) = the carrier of W1 /\ the carrier of W2 by Def2;

hence W1 /\ W2 = W1 by A1, RLSUB_1:30, XBOOLE_1:28; :: thesis: verum

end;then A1: the carrier of W1 c= the carrier of W2 by RLSUB_1:def 2;

the carrier of (W1 /\ W2) = the carrier of W1 /\ the carrier of W2 by Def2;

hence W1 /\ W2 = W1 by A1, RLSUB_1:30, XBOOLE_1:28; :: thesis: verum