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 a,b being Element of COMPLEX,x,y,z being FinSequence of COMPLEX st
  len x=len y & len y=len z holds |((a*x+b*y), z)| = a*|(x,z)| + b*|(y,z)|
proof
  let a,b be Element of COMPLEX,x,y,z be FinSequence of COMPLEX;
  assume that
A1: len x=len y and
A2: len y=len z;
  len (a*x)=len x & len (b*y)=len y by Th3;
  then |(a*x + b*y, z)| = |(a*x, z)| + |(b*y, z)| by A1,A2,Th55
    .= a*|(x,z)| + |(b*y,z)| by A1,A2,Th67
    .= a*|(x,z)| + b*|(y,z)| by A2,Th67;
  hence thesis;
end;
