reserve X,Y,Z,x,y,z for set;
reserve T,R for Tolerance of X;

theorem Th10:
  for R be Relation of X,Y st R is symmetric holds R |_2 Z is symmetric
proof
  let R be Relation of X,Y;
  assume R is symmetric;
  then
A1: R is_symmetric_in field R;
  now
    let x,y be object;
    assume that
A2: x in field(R|_2 Z) & y in field(R|_2 Z) and
A3: [x,y] in R|_2 Z;
A4: [x,y] in R by A3,XBOOLE_0:def 4;
A5: [y,x] in [:Z,Z:] by A3,ZFMISC_1:88;
    x in field R & y in field R by A2,WELLORD1:12;
    then [y,x] in R by A1,A4;
    hence [y,x] in R|_2 Z by A5,XBOOLE_0:def 4;
  end;
  then R|_2 Z is_symmetric_in field(R|_2 Z);
  hence thesis;
end;
