 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;
reserve G for non empty multMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;

theorem
  A * (B /\ C) c= (A * B) /\ (A * C)
proof
  let x be object;
  assume x in A * (B /\ C);
  then consider g1,g2 such that
A1: x = g1 * g2 & g1 in A and
A2: g2 in B /\ C;
  g2 in C by A2,XBOOLE_0:def 4;
  then
A3: x in A * C by A1;
  g2 in B by A2,XBOOLE_0:def 4;
  then x in A * B by A1;
  hence thesis by A3,XBOOLE_0:def 4;
end;
