reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem Th23:
  f c= g implies disjoin f c= disjoin g
proof
  assume
A1: f c= g;
  then
A2: dom f c= dom g by GRFUNC_1:2;
A3: dom disjoin f = dom f by Def3;
A4: dom disjoin g = dom g by Def3;
  now
    let x be object;
    assume
A5: x in dom disjoin f;
    then
A6: (disjoin f).x = [:f.x,{x}:] by A3,Def3;
    f.x = g.x by A1,A3,A5,GRFUNC_1:2;
    hence (disjoin f).x = (disjoin g).x by A2,A3,A5,A6,Def3;
  end;
  hence thesis by A2,A3,A4,GRFUNC_1:2;
end;
