reserve a,b,r for non unit non zero Real;
reserve X for non empty set,
        x for Tuple of 4,X;

theorem Th02:
  4-tuples_on X = the set of all <* d1, d2, d3, d4 *> where
    d1,d2,d3,d4 is Element of X
  proof
    hereby
      let x be object;
      assume x in 4-tuples_on X;
      then consider s be Element of X* such that
A1:   x = s and
A2:   len s = 4;
      1 in Seg 4 & 2 in Seg 4 & 3 in Seg 4 & 4 in Seg 4;
      then 1 in dom s & 2 in dom s & 3 in dom s & 4 in dom s
        by A2,FINSEQ_1:def 3;
      then reconsider d19 = s.1, d29 = s.2, d39 = s.3,
      d49 = s.4 as Element of X by FINSEQ_2:11;
      s = <* d19, d29, d39, d49 *> by A2,FINSEQ_4:76;
      hence x in the set of all <* d1, d2, d3, d4 *>
        where d1,d2,d3,d4 is Element of X by A1;
    end;
    let x be object;
    assume x in the set of all <* d1, d2, d3, d4 *>
      where d1,d2,d3,d4 is Element of X;
    then consider d1, d2, d3, d4 be Element of X such that
A3: x = <* d1, d2, d3,d4 *>;
    len <* d1, d2, d3, d4 *> = 4 & <* d1, d2, d3, d4 *> is Element of X*
      by FINSEQ_1:def 11,FINSEQ_4:76;
    hence x in 4-tuples_on X by A3;
  end;
