
theorem Th4:
  for G being Group, E being non empty set, x,y being Element of E,
  T being LeftOperation of G, E st x,y are_conjugated_under T holds y,x
  are_conjugated_under T
proof
  let G be Group;
  let E be non empty set;
  let x,y be Element of E;
  let T be LeftOperation of G, E;
  assume x,y are_conjugated_under T;
  then consider g be Element of G such that
A1: y = (T^g).x;
  x in E;
  then x in dom(T^g) by FUNCT_2:def 1;
  then (T^g").y = ((T^g")*(T^g)).x by A1,FUNCT_1:13
    .= (T^(g"*g)).x by Def1
    .= (T^1_G).x by GROUP_1:def 5
    .= (id E).x by Def1
    .= x;
  hence thesis;
end;
