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
  for x,y being FinSequence of COMPLEX st len x=len y holds
  |(x-y, x-y)| = |(x, x)| - 2*Re(|(x, y)|) + |(y, y)|
proof
  let x,y be FinSequence of COMPLEX;
  set z=|(x, y)|;
  assume len x=len y;
  then |(x-y, x-y)| =|(x, x)| - |(x, y)| - |(y, x)| + |(y, y)| by Th63
    .=|(x, x)| - |(x, y)| - (|(x, y)|)*' + |(y, y)| by Th64
    .=|(x, x)| - (z + z*') + |(y, y)|
    .=|(x, x)| - 2*Re(|(x, y)|) + |(y, y)| by Th20;
  hence thesis;
end;
