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 Th91:
  abs(c*z) = |.c.|*(abs z)
proof
  now
    let j be Nat;
    reconsider w = j as Element of NAT by ORDINAL1:def 12;
    assume
A1: j in Seg n;
    then reconsider c9 = z.j, cc = (c*z).j as Element of COMPLEX by Th57;
    reconsider ac = (abs z).w as Real;
    thus (abs(c*z)).j = |.cc.| by A1,Th88
      .= |.c*c9.| by A1,Th81
      .= |.c.|*|.c9.| by COMPLEX1:65
      .= |.c.|*ac by A1,Th88
      .= (|.c.|*(abs z)).j by RVSUM_1:45;
  end;
  hence thesis by FINSEQ_2:119;
end;
