theorem :: STIRL2_1:50
  (for n st n in dom rF holds rF.n <= r) implies
   Sum rF <= len rF * r
proof
  set L= len rF-->r;
  assume A1:n in dom rF implies rF.n <= r;
  A2:len L=len rF by FUNCOP_1:13;
  now let n;assume n in dom rF;
     then rF.n <= r & L.n = r by A1,FUNCOP_1:7;
     hence rF.n <= L.n;
  end;
  then Sum rF <= Sum L by Th56,A2;
  hence thesis by Th57;
end;
