reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;
reserve k,n for Nat,
  r,r9,r1,r2 for Real,
  c,c9,c1,c2,c3 for Element of COMPLEX;
reserve z,z1,z2 for FinSequence of COMPLEX;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
  A,B for Subset of COMPLEX n;

theorem Th58:
  k in Seg n & c1 = z1.k & c2 = z2.k implies (z1 + z2).k = c1 + c2
proof
  assume that
A1: k in Seg n and
A2: c1 = z1.k & c2 = z2.k;
  k in dom(z1+z2) by A1,Lm4;
  hence (z1 + z2).k = addcomplex.(c1,c2) by A2,FUNCOP_1:22
    .= c1 + c2 by BINOP_2:def 3;
end;
