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 Th81:
  k in Seg n & c9 = z.k implies (c*z).k = c*c9
proof
  assume that
A1: k in Seg n and
A2: c9 = z.k;
  k in dom(c*z) by A1,Lm4;
  hence (c*z).k = (c multcomplex).c9 by A2,FUNCT_1:12
    .= c*c9 by Lm3;
end;
