
theorem
for X be non empty set, F be FinSequence of X, G be Function of NAT,X
 st (for i be Nat holds F.i = G.i)
 holds F is disjoint_valued iff G is disjoint_valued
proof
   let X be non empty set, F be FinSequence of X, G be Function of NAT,X;
   assume A1: for i be Nat holds F.i = G.i;
   hereby assume
A2: F is disjoint_valued;
    now let x,y be object;
     assume A3: x <> y;
     per cases;
     suppose x in dom F & y in dom F; then
      G.x = F.x & G.y = F.y by A1;
      hence G.x misses G.y by A2,A3,PROB_2:def 2;
     end;
     suppose not x in dom F & x in dom G; then
      F.x = {} & G.x = F.x by A1,FUNCT_1:def 2;
      hence G.x misses G.y by XBOOLE_1:65;
     end;
     suppose not x in dom F & not x in dom G; then
      G.x = {} by FUNCT_1:def 2;
      hence G.x misses G.y by XBOOLE_1:65;
     end;
     suppose not y in dom F & y in dom G; then
      F.y = {} & G.y = F.y by A1,FUNCT_1:def 2;
      hence G.x misses G.y by XBOOLE_1:65;
     end;
     suppose not y in dom F & not y in dom G; then
      G.y = {} by FUNCT_1:def 2;
      hence G.x misses G.y by XBOOLE_1:65;
     end;
    end;
    hence G is disjoint_valued by PROB_2:def 2;
   end;
   assume A8: G is disjoint_valued;
   now let x,y be object;
    assume A9: x <> y;
    per cases;
    suppose x in dom G & y in dom G; then
     F.x = G.x & F.y = G.y by A1;
     hence F.x misses F.y by A8,A9,PROB_2:def 2;
    end;
    suppose A10: not x in dom G or not y in dom G;
     dom F c= NAT; then
     dom F c= dom G by FUNCT_2:def 1; then
     not x in dom F or not y in dom F by A10; then
     F.x = {} or F.y = {} by FUNCT_1:def 2;
     hence F.x misses F.y by XBOOLE_1:65;
    end;
   end;
   hence F is disjoint_valued by PROB_2:def 2;
end;
