
theorem Th17:
  for X be non-empty 1-element FinSequence,
      S be FieldFamily of X holds
    the set of all product <*s*> where s is Element of S.1 is
      Field_Subset of the set of all <*x*> where x is Element of X.1
proof
   let X be non-empty 1-element FinSequence, S be FieldFamily of X;
   set S1 = the set of all product <*s*> where s is Element of S.1;
   set X1 = the set of all <*x*> where x is Element of X.1;
   dom X = Seg 1 by FINSEQ_1:89; then
A1:len X = 1 by FINSEQ_1:def 3;
A2: 1 in Seg 1; then
   1 in dom X by FINSEQ_1:89; then
A4:X1 = product <*X.1*> by SRINGS_4:24;
A5:X = <*X.1*> by A1,FINSEQ_1:40;
   reconsider F = S.1 as Field_Subset of X.1 by A2,Def3;
   F is non empty;
   then consider s be object such that
A7: s in F;
   reconsider s as Element of S.1 by A7;
   S1 = SemiringProduct(S) by SRINGS_4:25; then
   reconsider S1 as Subset-Family of product X by SRINGS_4:22;
   now let A be set;
    assume A in S1; then
    consider s be Element of S.1 such that
A8:  A = product <*s*>;
    (X.1) \ s in F by MEASURE1:def 1; then
    product <* (X.1) \ s *> in S1;
    hence (product X) \ A in S1 by A5,A8,SRINGS_4:27;
   end;
   hence thesis by A4,A5,Th12,MEASURE1:def 1;
end;
