let V be Z_Module; :: thesis: for W1 being Submodule of V
for W2, W3 being strict Submodule of V st W1 is Submodule of W2 holds
W1 + W3 is Submodule of W2 + W3
let W1 be Submodule of V; :: thesis: for W2, W3 being strict Submodule of V st W1 is Submodule of W2 holds
W1 + W3 is Submodule of W2 + W3
let W2, W3 be strict Submodule of V; :: thesis: ( W1 is Submodule of W2 implies W1 + W3 is Submodule of W2 + W3 )
assume A1: W1 is Submodule of W2 ; :: thesis: W1 + W3 is Submodule of W2 + W3
(W1 + W3) + (W2 + W3) = ((W1 + W3) + W3) + W2 by Th96
.= (W1 + (W3 + W3)) + W2 by Th96
.= (W1 + W3) + W2 by Lm7
.= (W1 + W2) + W3 by Th96
.= W2 + W3 by A1, Th98 ;
hence W1 + W3 is Submodule of W2 + W3 by Th98; :: thesis: verum