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 Th107:
  for H1,H2 being strict Subgroup of G holds H1 in con_class H2
  iff H1,H2 are_conjugated
proof
  let H1,H2 be strict Subgroup of G;
  thus H1 in con_class H2 implies H1,H2 are_conjugated
  proof
    assume H1 in con_class H2;
    then
    ex H3 being strict Subgroup of G st H1 = H3 & H2,H3 are_conjugated by Def12
;
    hence thesis;
  end;
  assume H1,H2 are_conjugated;
  hence thesis by Def12;
end;
