
theorem Disjoint2:
  for x,y be set, F be FinSequence st F = <*x,y*> & x misses y holds
    F is disjoint_valued
proof
   let x,y be set, F be FinSequence;
   assume A1: F = <*x,y*> & x misses y;
   now let i,j be object;
    assume A6: i <> j;
    per cases;
    suppose not i in dom F or not j in dom F; then
     F.i = {} or F.j = {} by FUNCT_1:def 2;
     hence F.i misses F.j;
    end;
    suppose A7: i in dom F & j in dom F;
  len F = 2 & F.1 = x & F.2 = y by  A1,FINSEQ_1:44; then
     i in {1,2} & j in {1,2} by A7,FINSEQ_1:def 3,2; then
     (i = 1 or i = 2) & (j = 1 or j = 2) by TARSKI:def 2;
     hence F.i misses F.j by A1,A6;
    end;
   end;
   hence F is disjoint_valued by PROB_2:def 2;
end;
