 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;

theorem Th10:
  G is invertible iff for a,b ex r,l st a*r = b & l*a = b
proof
  thus G is invertible implies for a,b ex r,l st a*r = b & l*a = b
  proof
    assume
A1: for a,b ex r,l st op(G).(a,r) = b & op(G).(l,a) = b;
    let a,b;
    consider r, l such that
A2: op(G).(a,r) = b & op(G).(l,a) = b by A1;
    take r, l;
    thus thesis by A2;
  end;
  assume
A3: for a,b ex r,l st a*r = b & l*a = b;
  let a,b;
  consider r, l such that
A4: a*r = b & l*a = b by A3;
  take r, l;
  thus thesis by A4;
end;
