reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for addGroup-like non empty addMagma;
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 addGroup;
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;
reserve x,y,y1,y2 for set;
reserve G for addGroup;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem Th63:
  (H1 /\ H2) * a = (H1 * a) /\ (H2 * a)
proof
  let g;
  thus g in (H1 /\ H2) * a implies g in (H1 * a) /\ (H2 * a)
  proof
    assume g in (H1 /\ H2) * a;
    then consider h such that
A1: g = h * a and
A2: h in H1 /\ H2 by Th58;
    h in H2 by A2,Th82;
    then
A3: g in H2 * a by A1,Th58;
    h in H1 by A2,Th82;
    then g in H1 * a by A1,Th58;
    hence thesis by A3,Th82;
  end;
  assume
A4: g in (H1 * a) /\ (H2 * a);
  then g in H1 * a by Th82;
  then consider b such that
A5: g = b * a and
A6: b in H1 by Th58;
  g in H2 * a by A4,Th82;
  then consider c such that
A7: g = c * a and
A8: c in H2 by Th58;
  b = c by A5,A7,ThB16;
  then b in (H1 /\ H2) by A6,A8,Th82;
  hence thesis by A5,Th58;
end;
