 reserve m,n for Nat;
 reserve i,j for Integer;
 reserve S for non empty multMagma;
 reserve r,r1,r2,s,s1,s2,t for Element of S;
 reserve G for Group-like non empty multMagma;
 reserve e,h for Element of G;
 reserve G for Group;
 reserve f,g,h for Element of G;

theorem Th12:
  h * f = g iff f = h" * g
proof
  h * (h" * g) = h * h" * g by Def3
    .= 1_G * g by Def5
    .= g by Def4;
  hence h * f = g implies f = h" * g by Th6;
  assume f = h" * g;
  hence h * f = h * h" * g by Def3
    .= 1_G * g by Def5
    .= g by Def4;
end;
