 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;
reserve G for Group-like non empty multMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;
reserve H for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve G,G1,G2,G3 for Group;
reserve a,a1,a2,b,b1,b2,g,g1,g2 for Element of G;
reserve A,B for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve h,h1,h2 for Element of H;

theorem
  x in H * A iff ex g1,g2 st x = g1 * g2 & g1 in H & g2 in A
proof
  thus x in H * A implies ex g1,g2 st x = g1 * g2 & g1 in H & g2 in A
  proof
    assume x in H * A;
    then consider g1,g2 such that
A1: x = g1 * g2 and
A2: g1 in carr H and
A3: g2 in A;
    g1 in H by A2;
    hence thesis by A1,A3;
  end;
  given g1,g2 such that
A4: x = g1 * g2 and
A5: g1 in H and
A6: g2 in A;
  g1 in carr H by A5;
  hence thesis by A4,A6;
end;
