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 <> {} )
then reconsider x = the Element of A as Element of G by TARSKI:def 3;
assume A2: B <> {} ; :: thesis: A * B <> {}
then reconsider y = the Element of B as Element of G by TARSKI:def 3;
x * y in A * B by A1, A2;
hence A * B <> {} ; :: thesis: verum
end;
set x = the Element of A * B;
assume A * B <> {} ; :: thesis: ( A <> {} & B <> {} )
then ex a, b being Element of G st
( the Element of A * B = a * b & a in A & b in B ) by Th8;
hence ( A <> {} & B <> {} ) ; :: thesis: verum