
theorem Th4:
for X,A be set, f be PartFunc of X,REAL st f is nonpositive holds
  f|A is nonpositive
proof
    let X,A be set, f be PartFunc of X,REAL;
    assume A1: f is nonpositive;

    now let r be R_eal;
     assume r in rng(f|A); then
     consider x be object such that
A2:   x in dom(f|A) & r = (f|A).x by FUNCT_1:def 3;
     f.x <= 0 by A1,A2,MESFUNC6:53;
     hence r <= 0 by A2,FUNCT_1:47;
    end;
    hence f|A is nonpositive by MESFUNC5:def 1,def 2;
end;
