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 Th57:
  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>*x2)=len x2 by Th3;
  |(x1, -x2)| = |(x1, <i>*<i>*x2)| .= |(x1, <i>*(<i>*x2))| by Th44
    .= -<i>*(|(x1, <i>*x2)|) by A1,A2,Th50
    .= -<i>*(-<i>*(|(x1, x2)|)) by A1,Th50
    .= -(|(x1, x2)|);
  hence thesis;
end;
