
theorem Th6:
for X be non empty set, S be SigmaField of X, E1,E2 be Element of S
 st E1 misses E2 holds <* E1, E2 *> is Finite_Sep_Sequence of S
proof
   let X be non empty set, S be SigmaField of X, E1,E2 be Element of S;
   assume A0: E1 misses E2;
A2:dom <*E1,E2*> = {1,2} by FINSEQ_1:92;
   now let m,n be object;
    assume A3: m <> n;
    per cases;
    suppose m in dom <*E1,E2*> & n in dom <*E1,E2*>; then
     (m = 1 or m = 2) & (n = 1 or n = 2) by A2,TARSKI:def 2; then
     (<*E1,E2*>.m = E1 & <*E1,E2*>.n = E2)
       or (<*E1,E2*>.m = E2 & <*E1,E2*>.n = E1) by A3;
     hence <*E1,E2*>.m misses <*E1,E2*>.n by A0;
    end;
    suppose not m in dom <*E1,E2*> or not n in dom <*E1,E2*>; then
     <*E1,E2*>.m = {} or <*E1,E2*>.n = {} by FUNCT_1:def 2;
     hence <*E1,E2*>.m misses <*E1,E2*>.n;
    end;
   end; then
   <*E1,E2 *> is disjoint_valued;
   hence <*E1,E2*> is Finite_Sep_Sequence of S;
end;
