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 |(x,a*y+b*z)| = (a*')*|(x,y)| + (b*')*|(x,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*y)=len y & len (b*z)=len z by Th3;
  then |(x,a*y+b*z)| = |(x, a*y)| + |(x, b*z)| by A1,A2,Th60
    .= (a*')*|(x,y)| + |(x,b*z)| by A1,Th68
    .= (a*')*|(x,y)| + (b*')*|(x,z)| by A1,A2,Th68;
  hence thesis;
end;
