reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem Th37:
  for G being strict Group, H being strict Subgroup of G, a being
  Element of G holds H is maximal & not a in H implies gr(carr H \/ {a}) = G
proof
  let G be strict Group, H be strict Subgroup of G, a be Element of G;
  assume that
A1: H is maximal and
A2: not a in H;
  gr carr H is Subgroup of gr(carr H \/ {a}) by Th32,XBOOLE_1:7;
  then
A3: H is Subgroup of gr(carr H \/ {a}) by Th31;
  a in {a} by TARSKI:def 1;
  then a in carr H \/ {a} by XBOOLE_0:def 3;
  then H <> gr(carr H \/ {a}) by A2,Th29;
  hence thesis by A1,A3;
end;
