reserve i,j for Element of NAT,
  x,y,z for FinSequence of COMPLEX,
  c for Element of COMPLEX,
  R,R1,R2 for Element of i-tuples_on COMPLEX;
reserve C for Function of [:COMPLEX,COMPLEX:],COMPLEX;
reserve G for Function of [:REAL,REAL:],REAL;
reserve h for Function of COMPLEX,COMPLEX,
  g for Function of REAL,REAL;

theorem Th56:
  for x1,x2 being FinSequence of COMPLEX st len x1=len x2 holds
  |(-x1, x2)| = -|(x1, x2)|
proof
  let x1,x2 be FinSequence of COMPLEX;
  assume
A1: len x1=len x2;
A2: len (<i>*x1)=len x1 by Th3;
  |(-x1, x2)| = |(<i>*<i>*x1, x2)| .= |(<i>*(<i>*x1), x2)| by Th44
    .= <i>*(|(<i>*x1, x2)|) by A1,A2,Th49
    .= <i>*(<i>*(|(x1, x2)|)) by A1,Th49
    .= (-1)*|(x1, x2)|;
  hence thesis;
end;
