let G be non empty multMagma ; :: thesis: for A, B, C being Subset of G holds (A \/ B) * C = (A * C) \/ (B * C)
let A, B, C be Subset of G; :: thesis: (A \/ B) * C = (A * C) \/ (B * C)
thus (A \/ B) * C c= (A * C) \/ (B * C) :: according to XBOOLE_0:def 10 :: thesis: (A * C) \/ (B * C) c= (A \/ B) * C
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (A \/ B) * C or x in (A * C) \/ (B * C) )
assume x in (A \/ B) * C ; :: thesis: x in (A * C) \/ (B * C)
then consider g1, g2 being Element of G such that
A1: x = g1 * g2 and
A2: g1 in A \/ B and
A3: g2 in C ;
( g1 in A or g1 in B ) by ;
then ( x in A * C or x in B * C ) by A1, A3;
hence x in (A * C) \/ (B * C) by XBOOLE_0:def 3; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (A * C) \/ (B * C) or x in (A \/ B) * C )
assume A4: x in (A * C) \/ (B * C) ; :: thesis: x in (A \/ B) * C
now :: thesis: x in (A \/ B) * C
per cases ( x in A * C or x in B * C ) by ;
suppose x in A * C ; :: thesis: x in (A \/ B) * C
then consider g1, g2 being Element of G such that
A5: x = g1 * g2 and
A6: g1 in A and
A7: g2 in C ;
g1 in A \/ B by ;
hence x in (A \/ B) * C by A5, A7; :: thesis: verum
end;
suppose x in B * C ; :: thesis: x in (A \/ B) * C
then consider g1, g2 being Element of G such that
A8: x = g1 * g2 and
A9: g1 in B and
A10: g2 in C ;
g1 in A \/ B by ;
hence x in (A \/ B) * C by ; :: thesis: verum
end;
end;
end;
hence x in (A \/ B) * C ; :: thesis: verum