let G be non empty multMagma ; :: thesis: for A, B being Subset of G holds
( ( A <> {} & B <> {} ) iff A * B <> {} )
let A, B be Subset of G; :: thesis: ( ( A <> {} & B <> {} ) iff A * B <> {} )
thus ( A <> {} & B <> {} implies A * B <> {} ) :: thesis: ( A * B <> {} implies ( A <> {} & B <> {} ) )

proof

assume A1: A <> {} ; :: thesis: ( not B <> {} or A * B <> {} )
reconsider x = the Element of A as Element of G by A1, TARSKI:def 3;
assume A2: B <> {} ; :: thesis: A * B <> {}
reconsider y = the Element of B as Element of G by A2, TARSKI:def 3;
x * y in A * B by A1, A2;
hence A * B <> {} ; :: thesis: verum

end;

assume A3: A * B <> {} ; :: thesis: ( A <> {} & B <> {} )
consider x being Element of A * B;
ex a, b being Element of G st
( x = a * b & a in A & b in B ) by A3, Th12;
hence ( A <> {} & B <> {} ) ; :: thesis: verum