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

for W2, W3 being strict Subspace of V st W1 is Subspace of W2 holds

W1 + W3 is Subspace of W2 + W3

let W1 be Subspace of V; :: thesis: for W2, W3 being strict Subspace of V st W1 is Subspace of W2 holds

W1 + W3 is Subspace of W2 + W3

let W2, W3 be strict Subspace of V; :: thesis: ( W1 is Subspace of W2 implies W1 + W3 is Subspace of W2 + W3 )

assume A1: W1 is Subspace of W2 ; :: thesis: W1 + W3 is Subspace of W2 + W3

(W1 + W3) + (W2 + W3) = (W1 + W3) + (W3 + W2) by Lm1

.= ((W1 + W3) + W3) + W2 by Th6

.= (W1 + (W3 + W3)) + W2 by Th6

.= (W1 + W3) + W2 by Lm3

.= W1 + (W3 + W2) by Th6

.= W1 + (W2 + W3) by Lm1

.= (W1 + W2) + W3 by Th6

.= W2 + W3 by A1, Th8 ;

hence W1 + W3 is Subspace of W2 + W3 by Th8; :: thesis: verum

for W2, W3 being strict Subspace of V st W1 is Subspace of W2 holds

W1 + W3 is Subspace of W2 + W3

let W1 be Subspace of V; :: thesis: for W2, W3 being strict Subspace of V st W1 is Subspace of W2 holds

W1 + W3 is Subspace of W2 + W3

let W2, W3 be strict Subspace of V; :: thesis: ( W1 is Subspace of W2 implies W1 + W3 is Subspace of W2 + W3 )

assume A1: W1 is Subspace of W2 ; :: thesis: W1 + W3 is Subspace of W2 + W3

(W1 + W3) + (W2 + W3) = (W1 + W3) + (W3 + W2) by Lm1

.= ((W1 + W3) + W3) + W2 by Th6

.= (W1 + (W3 + W3)) + W2 by Th6

.= (W1 + W3) + W2 by Lm3

.= W1 + (W3 + W2) by Th6

.= W1 + (W2 + W3) by Lm1

.= (W1 + W2) + W3 by Th6

.= W2 + W3 by A1, Th8 ;

hence W1 + W3 is Subspace of W2 + W3 by Th8; :: thesis: verum