
theorem Th15:
  for L be non empty multMagma for p,q,r be sequence of L for t be
  FinSequence of 3-tuples_on NAT for P be Permutation of dom t for t1 be
  FinSequence of 3-tuples_on NAT st t1 = t*P holds prodTuples(p,q,r,t1) =
  prodTuples(p,q,r,t)*P
proof
  let L be non empty multMagma;
  let p,q,r be sequence of L;
  let t be FinSequence of 3-tuples_on NAT;
  let P be Permutation of dom t;
  let t1 be FinSequence of 3-tuples_on NAT;
A1: rng P = dom t by FUNCT_2:def 3;
  assume
A2: t1 = t*P;
  then
A3: dom P = dom t1 by A1,RELAT_1:27;
A4: now
    let x be object;
    assume
A5: x in dom t1;
    then reconsider i=x as Element of NAT;
A6: prodTuples(p,q,r,t1).i = (p.((t1/.i)/.1))*(q.((t1/.i)/.2))*(r.((t1/.i
)/.3)) & (prodTuples(p,q,r,t)*P).x = prodTuples(p,q,r,t).(P.x) by A3,A5,Def5,
FUNCT_1:13;
    reconsider j=P.i as Element of NAT;
A7: P.i in rng P by A3,A5,FUNCT_1:def 3;
    t1/.i = t1.i by A5,PARTFUN1:def 6
      .= t.(P.i) by A2,A5,FUNCT_1:12
      .= t/.j by A1,A7,PARTFUN1:def 6;
    hence prodTuples(p,q,r,t1).x = (prodTuples(p,q,r,t)*P).x by A1,A7,A6,Def5;
  end;
  len prodTuples(p,q,r,t1) = len t1 by Def5;
  then
A8: dom prodTuples(p,q,r,t1) = Seg len t1 by FINSEQ_1:def 3;
  len prodTuples(p,q,r,t) = len t by Def5;
  then rng P = dom prodTuples(p,q,r,t) by A1,FINSEQ_3:29;
  then
A9: dom (prodTuples(p,q,r,t)*P) = dom t1 by A3,RELAT_1:27;
  dom t1 = Seg len t1 by FINSEQ_1:def 3;
  hence thesis by A8,A9,A4,FUNCT_1:2;
end;
