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;

theorem
  H1 is Subgroup of H2 implies H1 /\ H3 is Subgroup of H2 /\ H3
proof
  assume H1 is Subgroup of H2;
  then (the carrier of H1) /\ (the carrier of H3) c= (the carrier of H2) /\ (
  the carrier of H3) by DefA5,XBOOLE_1:26;
  then
  the carrier of H1 /\ H3 c= (the carrier of H2) /\ (the carrier of H3) by Th80
;
  then the carrier of H1 /\ H3 c= the carrier of H2 /\ H3 by Th80;
  hence thesis by Th57;
end;
