reserve x,y for set,
  i for Nat;
reserve V for non empty CLSStruct,
  u,v,v1,v2,v3 for VECTOR of V,
  A for Subset of V,
  l, l1, l2 for C_Linear_Combination of A,
  x,y,y1,y2 for set,
  a,b for Complex,
  F for FinSequence of the carrier of V,
  f for Function of the carrier of V, COMPLEX;
reserve K,L,L1,L2,L3 for C_Linear_Combination of V;
reserve e,e1,e2 for Element of C_LinComb V;

theorem Th44:
  for a being Real , z being Complex st 0<=a & a<=1
  holds |.a*z.| = a*|.z.| & |.(1r-a)*z.| = (1r-a)*|.z.|
proof
  let a be Real;
  let z be Complex;
  assume that
A1: 0<=a and
A2: a<=1;
A3: |.(1r-a)*z.| = |.1r-a.|*|.z.| by COMPLEX1:65
    .= (1r-a)*|.z.| by A2,COMPLEX1:43,XREAL_1:48;
  |.a*z.| = |.a.|*|.z.| by COMPLEX1:65
    .= a * |.z.| by A1,COMPLEX1:43;
  hence thesis by A3;
end;
