reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];

theorem
  (.:R).:{_{X1}_} = {} implies R.:^(X1\/X2) = R.:^X2
proof
A1: {_{X1\/X2}_} = {_{X1}_} \/ {_{X2}_} by Th3;
  assume
A2: (.:R).:{_{X1}_} = {};
  R .:^ (X1\/X2) =
  Intersect((.:R).:{_{X1}_} \/ ((.:R).:{_{X2}_})) by A1,RELAT_1:120
    .= R.:^X2 by A2;
  hence thesis;
end;
