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 Th63:
  for x1,x2,y1,y2 being FinSequence of COMPLEX st
  len x1=len x2 & len x2=len y1 & len y1=len y2 holds
  |(x1-x2, y1-y2)| = |(x1, y1)| - |(x1, y2)| - |(x2, y1)| + |(x2, y2)|
proof
  let x1,x2,y1,y2 be FinSequence of COMPLEX;
  assume that
A1: len x1=len x2 and
A2: len x2=len y1 and
A3: len y1=len y2;
  |(x1,y1-y2)| = |(x1,y1)| - |(x1,y2)| by A1,A2,A3,Th61; then
A4: |(x1,y1-y2)| - |(x2,y1-y2)| = (|(x1,y1)|-|(x1,y2)|)-(|(x2,y1)|-|(x2,y2)|
  ) by A2,A3,Th61;
  len (y1 - y2)=len y1 by A3,Th7;
  hence thesis by A1,A2,A4,Th59;
end;
