
theorem
for F be FinSequence of ExtREAL, G be ExtREAL_sequence
 st (for i be Nat holds F.i = G.i)
 holds F is nonnegative iff G is nonnegative
proof
   let F be FinSequence of ExtREAL, G be ExtREAL_sequence;
   assume
A1: for i be Nat holds F.i = G.i;
   hereby assume A3: F is nonnegative;
    now let i be object;
     assume A4: i in dom G;
     per cases;
     suppose i in dom F; then
      G.i = F.i by A1;
      hence G.i >= 0 by A3,SUPINF_2:51;
     end;
     suppose not i in dom F; then
      F.i = 0 by FUNCT_1:def 2;
      hence G.i >= 0 by A1,A4;
     end;
    end;
    hence G is nonnegative by SUPINF_2:52;
   end;
   assume A5: G is nonnegative;
   now let i be object;
    per cases;
    suppose i in dom F; then
     F.i = G.i by A1;
     hence F.i >= 0 by A5,SUPINF_2:51;
    end;
    suppose not i in dom F;
     hence F.i >= 0 by FUNCT_1:def 2;
    end;
   end;
   hence F is nonnegative by SUPINF_2:51;
end;
