reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th46:
  for f,g be complex-functions-valued FinSequence-yielding FinSequence holds
    Sum (f^g) = (Sum f) ^ (Sum g)
proof
  let f,g be complex-functions-valued FinSequence-yielding FinSequence;
  A1:len (Sum f)=len f & len (Sum g)=len g & len (Sum (f^g))=len (f^g)
    by CARD_1:def 7;
  A2:len (f^g)=len f+len g & len ((Sum f) ^ (Sum g))= len f + len g
    by CARD_1:def 7,FINSEQ_1:22;
  A3:dom f= dom Sum f & dom g= dom Sum g by Def8;
  for i st 1<= i & i <= len f+len g holds (Sum (f^g)).i = ((Sum f) ^ (Sum g)).i
  proof
    let i such that A4: 1<= i & i <= len f+len g;
    A5:(Sum (f^g)).i = Sum ((f^g).i) & (Sum f).i= Sum (f.i) by Def8;
    A6:i in dom (f^g) by A4,A2,FINSEQ_3:25;
    per cases by A6,FINSEQ_1:25;
    suppose i in dom f;
      then (f^g).i = f.i & (Sum f^Sum g).i = (Sum f).i by A3,FINSEQ_1:def 7;
      hence thesis by A5;
    end;
    suppose ex j st j in dom g & i=len f+j;
      then consider j such that
      A7:j in dom g & i=len f+j;
      (f^g).i = g.j & (Sum f^Sum g).i = (Sum g).j by A7,A3,A1,FINSEQ_1:def 7;
      hence thesis by A5,Def8;
    end;
  end;
  hence thesis by A1,A2;
end;
