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;
reserve x,y,x1,u,w for Nat;

theorem Th21:
  c1 + (n|->c2) = n|->(c1+c2)
proof
A1: len (c1+(n|->c2)) = len (n|->c2)=n=len (n|->(c1+c2)) by CARD_1:def 7;
  now let i such that
A2:   1<= i <= n;
A3:   i in dom (c1 + (n|->c2)) by A2,A1,FINSEQ_3:25;
A4:   i in Seg n by A2;
    hence (n|->(c1+c2)).i = c1+c2 by FINSEQ_2:57
      .= c1 + ((n|->c2).i) by A4,FINSEQ_2:57
      .= (c1 + (n|->c2)).i by A3,VALUED_1:def 2;
   end;
   hence thesis by FINSEQ_1:14,A1;
end;
