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 Th20:
  for PA,PB being a_partition of Y holds PA '<' PB iff ERl(PA) c= ERl(PB)
proof
  let PA,PB be a_partition of Y;
  set RA = ERl PA, RB = ERl PB;
  hereby
    assume
A1: PA '<' PB;
 for x1,x2 being object holds [x1,x2] in RA implies [x1,x2] in RB
    proof
      let x1,x2 be object;
      assume [x1,x2] in RA;
      then consider A being Subset of Y such that
A2:   A in PA and
A3:   x1 in A & x2 in A by Def6;
   ex y st y in PB & A c= y by A1,A2,SETFAM_1:def 2;
      hence thesis by A3,Def6;
    end;
    hence ERl(PA) c= ERl(PB);
  end;
  assume
A4: ERl(PA) c= ERl(PB);
 for x st x in PA ex y st y in PB & x c= y
  proof
    let x;
    assume
A5: x in PA;
then A6: x<>{} by EQREL_1:def 4;
    set x0 = the Element of x;
    set y={z where z is Element of Y:[x0,z] in ERl(PB)};
A7: x c= y
    proof
      let x1 be object;
      assume
A8:  x1 in x;
then   [x0,x1] in RA by A5,Def6;
      hence thesis by A4,A5,A8;
    end;
    set x1 = the Element of x;
 [x0,x1] in RA by A5,A6,Def6;
    then consider B being Subset of Y such that
A9: B in PB and
A10: x0 in B and x1 in B by A4,Def6;
A11: y c= B
    proof
      let x2 be object;
      assume x2 in y;
then   ex z being Element of Y st z=x2 & [x0,z] in ERl(PB);
      then consider B2 being Subset of Y such that
A12:  B2 in PB and
A13:  x0 in B2 and
A14:  x2 in B2 by Def6;
  B2 meets B by A10,A13,XBOOLE_0:3;
      hence thesis by A9,A12,A14,EQREL_1:def 4;
    end;
 B c= y
    proof
      let x2 be object;
      assume
A15:  x2 in B;
then   [x0,x2] in RB by A9,A10,Def6;
      hence thesis by A15;
    end;
then  y=B by A11,XBOOLE_0:def 10;
    hence thesis by A7,A9;
  end;
  hence thesis by SETFAM_1:def 2;
end;
