reserve i,j,k,n,m for Nat,
  x,y,z,y1,y2 for object, X,Y,D for set,
  p,q for XFinSequence;
reserve k1,k2 for Nat;
reserve D for non empty set,
  F,G for XFinSequence of D,
  b for BinOp of D,
  d,d1,d2 for Element of D;
reserve F for XFinSequence,
        rF,rF1,rF2 for real-valued XFinSequence,
        r for Real,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

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;
