
theorem Th1:
for X be non empty set, A be set, r be Real holds r(#)chi(A,X) = chi(r,A,X)
proof
   let X be non empty set, A be set, r be Real;
   for x be Element of X holds (r(#)chi(A,X)).x = chi(r,A,X).x
   proof
    let x be Element of X;
    x in X; then
    x in dom(r(#)chi(A,X)) by FUNCT_2:def 1; then
A2: (r(#)chi(A,X)).x = r * chi(A,X).x by MESFUNC1:def 6;
    per cases;
    suppose x in A; then
     chi(A,X).x = 1 & chi(r,A,X).x = r by Def1,FUNCT_3:def 3;
     hence (r(#)chi(A,X)).x = chi(r,A,X).x by A2,XXREAL_3:81;
    end;
    suppose not x in A; then
     chi(A,X).x = 0 & chi(r,A,X).x = 0 by Def1,FUNCT_3:def 3;
     hence (r(#)chi(A,X)).x = chi(r,A,X).x by A2;
    end;
   end;
   hence thesis by FUNCT_2:def 8;
end;
