reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;

theorem
  EqR1 \/ EqR2 = nabla X implies EqR1 = nabla X or EqR2 = nabla X
proof
  assume
A1: EqR1 \/ EqR2 = nabla X;
  X <> {} implies EqR1 = nabla X or EqR2 = nabla X
  proof
    set y = the Element of X;
A2: now
      let x;
      assume
A3:   x in X;
      then
      Class(nabla X,x) = Class(EqR1,x) or Class(nabla X,x) = Class(EqR2,x)
      by A1,Th29;
      hence Class(EqR1,x) = X or Class(EqR2,x) = X by A3,Th26;
    end;
    assume X <> {};
    then Class(EqR1,y) = X or Class(EqR2,y) = X by A2;
    hence thesis by Th27;
  end;
  hence thesis;
end;
