 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
  for G being finite Group, H being Subgroup of G holds card G = card H
  implies the multMagma of H = the multMagma of G
proof
  let G be finite Group, H be Subgroup of G;
  assume
A1: card G = card H;
A2: the carrier of H c= the carrier of G by Def5;
  the carrier of H = the carrier of G
  proof
    assume the carrier of H <> the carrier of G;
    then the carrier of H c< the carrier of G by A2;
    hence thesis by A1,CARD_2:48;
  end;
  hence thesis by Th61;
end;
