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
  0c*z = 0c n
proof
A1: rng z c= COMPLEX;
  thus 0c*z = multcomplex[;](0c,(id COMPLEX)*z) by FUNCOP_1:34
    .= multcomplex[;](0c,z) by A1,RELAT_1:53
    .= 0c n by Th51,Th54,BINOP_2:1,FINSEQOP:76;
end;
