
theorem Th5:
  for S being unital non empty multMagma, E being non empty set,
  x,y,z being Element of E, T being LeftOperation of S, E st x,y
  are_conjugated_under T & y,z are_conjugated_under T holds x,z
  are_conjugated_under T
proof
  let S be unital non empty multMagma;
  let E be non empty set;
  let x,y,z be Element of E;
  let T be LeftOperation of S, E;
  assume x,y are_conjugated_under T;
  then consider s1 be Element of S such that
A1: y = (T^s1).x;
  assume y,z are_conjugated_under T;
  then consider s2 be Element of S such that
A2: z = (T^s2).y;
  x in E;
  then x in dom(T^s1) by FUNCT_2:def 1;
  then z = ((T^s2)*(T^s1)).x by A1,A2,FUNCT_1:13
    .= (T^(s2*s1)).x by Def1;
  hence thesis;
end;
