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
  (c1 + c2)*z = c1*z + c2*z
proof
  set c1M = multcomplex[;](c1,id COMPLEX), c2M = multcomplex[;](c2,id COMPLEX);
  thus (c1 + c2)*z = multcomplex[;](addcomplex.(c1,c2),id COMPLEX)*z by
BINOP_2:def 3
    .= addcomplex.:(c1M,c2M)*z by Th54,FINSEQOP:35
    .= c1*z + c2*z by FUNCOP_1:25;
end;
