reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem Th34:
  ERl(%I(Y)) = id Y
proof
A1: union %I(Y)=Y by EQREL_1:def 4;
A2: ERl(%I(Y)) c= id Y
  proof
    let x1,x2 be object;
    assume [x1,x2] in ERl(%I(Y));
    then consider a being Subset of Y such that
A3: a in %I(Y) and
A4: x1 in a & x2 in a by Def6;
     %I(Y) = the set of all {x} where x is Element of Y by EQREL_1:37;
    then consider x be Element of Y such that
A5: a={x} by A3;
    x1=x & x2=x by A4,A5,TARSKI:def 1;
    hence thesis by RELAT_1:def 10;
  end;
  id Y c= ERl(%I(Y))
  proof
    let x1,x2 be object;
    assume
A6: [x1,x2] in id Y; then
A7: x1 in Y by RELAT_1:def 10;
A8: x1=x2 by A6,RELAT_1:def 10;
    ex y being set st x1 in y & y in %I(Y) by A1,A7,TARSKI:def 4;
    hence thesis by A8,Def6;
  end;
  hence thesis by A2;
end;
