 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;
reserve M for non empty multLoopStr;
reserve H for non empty SubStr of G,
  N for non empty MonoidalSubStr of G;

theorem
  for n1,n2 being Element of NAT, m1,m2 being Element of <NAT,*,1> st
    n1 = m1 & n2 = m2 holds m1*m2 = n1*n2
proof
  let n1,n2 be Element of NAT, m1,m2 be Element of <NAT,*,1>;
  the multMagma of <NAT,*,1> = <NAT,*> by Def22;
  then reconsider x1 = m1, x2 = m2 as Element of <NAT,*>;
  x1*x2 = m1*m2 by Th18;
  hence thesis by Th50;
end;
