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

for W2 being strict Subspace of V holds

( W1 is Subspace of W2 iff W1 + W2 = W2 )

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

( W1 is Subspace of W2 iff W1 + W2 = W2 )

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

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

for W2 being strict Subspace of V holds

( W1 is Subspace of W2 iff W1 + W2 = W2 )

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

( W1 is Subspace of W2 iff W1 + W2 = W2 )

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

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

proof

thus
( W1 + W2 = W2 implies W1 is Subspace of W2 )
by Th7; :: thesis: verum
assume
W1 is Subspace of W2
; :: thesis: W1 + W2 = W2

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

hence W1 + W2 = W2 by Lm3; :: thesis: verum

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

hence W1 + W2 = W2 by Lm3; :: thesis: verum