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 Th59:
  for x1,x2,x3 being FinSequence of COMPLEX st
  len x1=len x2 & len x2=len x3 holds
  |(x1-x2, x3)| = |(x1, x3)| - |(x2, x3)|
proof
  let x1,x2,x3 be FinSequence of COMPLEX;
  assume that
A1: len x1=len x2 and
A2: len x2=len x3;
  len (-x2)=len x2 by Th5;
  then |(x1 - x2, x3)| = |(x1, x3)| + |(-x2, x3)| by A1,A2,Th55
    .= |(x1, x3)| + - |(x2, x3)| by A2,Th56;
  hence thesis;
end;
