
theorem Th38:
for X be non empty set, F be Functional_Sequence of X,ExtREAL,
 x be Element of X holds (-F)#x = -(F#x)
proof
    let X be non empty set, F be Functional_Sequence of X,ExtREAL,
    x be Element of X;
    now let n be Element of NAT;
A1:  dom(-(F#x)) = NAT by FUNCT_2:def 1;
A2:  ((-F)#x).n = ((-F).n).x by MESFUNC5:def 13
      .= (-(F.n)).x by Th37;
A3:  (-(F#x)).n = -((F#x).n) by A1,MESFUNC1:def 7
      .= -((F.n).x) by MESFUNC5:def 13;
A4:  dom(F.n) = dom(-(F.n)) by MESFUNC1:def 7;
     per cases;
     suppose x in dom(F.n);
      hence ((-F)#x).n = (-(F#x)).n by A3,A2,A4,MESFUNC1:def 7;
     end;
     suppose A5: not x in dom(F.n); then
A6:   not x in dom(-(F.n)) by MESFUNC1:def 7;
      (-(F#x)).n = -0 by A3,A5,FUNCT_1:def 2;
      hence ((-F)#x).n = (-(F#x)).n by A6,A2,FUNCT_1:def 2;
     end;
    end;
    hence (-F)#x = -(F#x) by FUNCT_2:def 8;
end;
