let G be non empty multMagma ; :: thesis: for A, B, C being Subset of G holds (A /\ B) * C c= (A * C) /\ (B * C)

let A, B, C be Subset of G; :: thesis: (A /\ B) * C c= (A * C) /\ (B * C)

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 B by A2, XBOOLE_0:def 4;

then A4: x in B * C by A1, A3;

g1 in A by A2, XBOOLE_0:def 4;

then x in A * C by A1, A3;

hence x in (A * C) /\ (B * C) by A4, XBOOLE_0:def 4; :: thesis: verum

let A, B, C be Subset of G; :: thesis: (A /\ B) * C c= (A * C) /\ (B * C)

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 B by A2, XBOOLE_0:def 4;

then A4: x in B * C by A1, A3;

g1 in A by A2, XBOOLE_0:def 4;

then x in A * C by A1, A3;

hence x in (A * C) /\ (B * C) by A4, XBOOLE_0:def 4; :: thesis: verum