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 Th80:
  (for H being Subgroup of G st H = H1 /\ H2 holds the carrier of
  H = (the carrier of H1) /\ (the carrier of H2)) & for H being strict Subgroup
  of G holds the carrier of H = (the carrier of H1) /\ (the carrier of H2)
  implies H = H1 /\ H2
proof
A1: the carrier of H1 = carr(H1) & the carrier of H2 = carr(H2);
  thus for H being Subgroup of G st H = H1 /\ H2 holds the carrier of H = (the
  carrier of H1) /\ (the carrier of H2)
  proof
    let H be Subgroup of G;
    assume H = H1 /\ H2;
    hence the carrier of H = carr(H1) /\ carr(H2) by Def10
      .= (the carrier of H1) /\ (the carrier of H2);
  end;
  let H be strict Subgroup of G;
  assume the carrier of H = (the carrier of H1) /\ (the carrier of H2);
  hence thesis by A1,Def10;
end;
