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;
reserve F,G for Cardinal-Function;

theorem
  dom F = dom G & (for x st x in dom F holds F.x c= G.x) implies Sum F c= Sum
  G
proof
  assume that
A1: dom F = dom G and
A2: for x st x in dom F holds F.x c= G.x;
  Union disjoin F c= Union disjoin G
  proof
    let x be object;
    assume x in Union disjoin F;
    then consider X such that
A3: x in X and
A4: X in rng disjoin F by TARSKI:def 4;
    consider y being object such that
A5: y in dom disjoin F and
A6: X = (disjoin F).y by A4,FUNCT_1:def 3;
A7: y in dom F by A5,Def3;
    then
A8: F.y c= G.y by A2;
A9: X = [:F.y,{y}:] by A6,A7,Def3;
A10: (disjoin G).y = [:G.y,{y}:] by A1,A7,Def3;
A11: y in dom disjoin G by A1,A7,Def3;
A12: X c= [:G.y,{y}:] by A8,A9,ZFMISC_1:95;
    [:G.y,{y}:] in rng disjoin G by A10,A11,FUNCT_1:def 3;
    hence thesis by A3,A12,TARSKI:def 4;
  end;
  hence thesis by CARD_1:11;
end;
