reserve i,j,n,n1,n2,m,k,l,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat,
        F for XFinSequence,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

theorem Th8:
  c+ (cF1^cF2) = (c+cF1) ^ (c+cF2)
proof
A1: dom (c+cF1) =dom cF1 & dom (c+cF2) =dom cF2 &
  dom (c+ (cF1^cF2)) = dom (cF1^cF2) by VALUED_1:def 2;
A2: len (cF1^cF2) = len cF1+len cF2 &
    len ((c+cF1)^(c+cF2)) = len (c+cF1)+len (c+cF2) by AFINSQ_1:17;
  for x be object st x in dom (c+ (cF1^cF2)) holds
    (c+ (cF1^cF2)).x = ((c+cF1) ^ (c+cF2)).x
  proof
    let x be object;
    assume
A3:   x in dom (c+ (cF1^cF2));
    then reconsider i=x as Nat;
    per cases by A3,A1,AFINSQ_1:20;
    suppose
A4:     i in dom cF1;
      then
A5:    (cF1^cF2).i = cF1.i & ((c+cF1) ^ (c+cF2)).i = (c+cF1).i
        by A1,AFINSQ_1:def 3;
      hence (c+ (cF1^cF2)).x = c+ (cF1.i) by A3,VALUED_1:def 2
        .= ((c+cF1)^(c+cF2)).x by A5,A4,A1,VALUED_1:def 2;
    end;
    suppose ex n st n in dom cF2 & i= len cF1 +n;
      then consider n such that
A6:     n in dom cF2 & i= len cF1 +n;
A7:    (cF1^cF2).i = cF2.n & ((c+cF1) ^ (c+cF2)).i = (c+cF2).n
        by A1,A6,AFINSQ_1:def 3;
      hence (c+ (cF1^cF2)).x = c +(cF2.n) by A3,VALUED_1:def 2
        .= ((c+cF1) ^ (c+cF2)).x by A1,A6,A7,VALUED_1:def 2;
    end;
  end;
hence thesis by A1,A2,FUNCT_1:2;
end;
