
theorem Th17:
for X be set, A be Subset of X, er be ExtReal holds
  (er >= 0 implies chi(er,A,X) is nonnegative)
& (er <= 0 implies chi(er,A,X) is nonpositive)
proof
   let X be set, A be Subset of X, er be ExtReal;
   hereby assume a1: er >= 0;
    now let x be object;
     assume a2: x in dom chi(er,A,X);
     x in A implies chi(er,A,X).x >= 0 by a1,Def1;
     hence chi(er,A,X).x >= 0 by a2,Def1;
    end;
    hence chi(er,A,X) is nonnegative by SUPINF_2:52;
   end;
   assume a3: er <= 0;
   now let x be set;
    assume a4: x in dom chi(er,A,X);
    x in A implies chi(er,A,X).x <= 0 by a3,Def1;
    hence chi(er,A,X).x <= 0 by a4,Def1;
   end;
   hence chi(er,A,X) is nonpositive by MESFUNC5:9;
end;
