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;

theorem Th54:
  multcomplex is_distributive_wrt addcomplex
proof
  now
    let c1,c2,c3;
    thus multcomplex.(c1,addcomplex.(c2,c3)) = multcomplex.(c1,c2+c3) by
BINOP_2:def 3
      .= c1*(c2+c3) by BINOP_2:def 5
      .= c1*c2+c1*c3
      .= addcomplex.(c1*c2,c1*c3) by BINOP_2:def 3
      .= addcomplex.(multcomplex.(c1,c2),c1*c3) by BINOP_2:def 5
      .= addcomplex.(multcomplex.(c1,c2),multcomplex.(c1,c3)) by BINOP_2:def 5;
    thus multcomplex.(addcomplex.(c1,c2),c3) = multcomplex.(c1+c2,c3) by
BINOP_2:def 3
      .= (c1+c2)*c3 by BINOP_2:def 5
      .= c1*c3+c2*c3
      .= addcomplex.(c1*c3,c2*c3) by BINOP_2:def 3
      .= addcomplex.(multcomplex.(c1,c3),c2*c3) by BINOP_2:def 5
      .= addcomplex.(multcomplex.(c1,c3),multcomplex.(c2,c3)) by BINOP_2:def 5;
  end;
  hence thesis by BINOP_1:11;
end;
