reserve x,y,y1,y2 for set;
reserve G for Group;
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;
reserve L for Subset of Subgroups G;

theorem
  for H1,H2 being strict Subgroup of G holds con_class H1 = con_class H2
  iff con_class H1 meets con_class H2
proof
  let H1,H2 be strict Subgroup of G;
  thus con_class H1 = con_class H2 implies con_class H1 meets con_class H2
  by Th109;
  assume con_class H1 meets con_class H2;
  then consider x being object such that
A1: x in con_class H1 and
A2: x in con_class H2 by XBOOLE_0:3;
  reconsider x as strict Subgroup of G by A1,Th106;
A3: H1,x are_conjugated by A1,Th107;
  thus con_class H1 c= con_class H2
  proof
    let y be object;
    assume y in con_class H1;
    then consider H3 being strict Subgroup of G such that
A4: H3 = y and
A5: H1,H3 are_conjugated by Def12;
A6: H2,x are_conjugated by A2,Th107;
    x,H1 are_conjugated by A1,Th107;
    then x,H3 are_conjugated by A5,Th105;
    then H2,H3 are_conjugated by A6,Th105;
    hence thesis by A4,Def12;
  end;
  let y be object;
  assume y in con_class H2;
  then consider H3 being strict Subgroup of G such that
A7: H3 = y and
A8: H2,H3 are_conjugated by Def12;
  x,H2 are_conjugated by A2,Th107;
  then x,H3 are_conjugated by A8,Th105;
  then H1,H3 are_conjugated by A3,Th105;
  hence thesis by A7,Def12;
end;
