 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 Th53:
  G is commutative Group implies H is commutative
proof
  assume
A1: G is commutative Group;
  thus for h1,h2 holds h1 * h2 = h2 * h1
  proof
    let h1,h2;
    reconsider g1 = h1, g2 = h2 as Element of G by Th42;
    thus h1 * h2 = g1 * g2 by Th43
      .= g2 * g1 by A1,Lm2
      .= h2 * h1 by Th43;
  end;
end;
