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;

theorem
  for x1,y1 being FinSequence of COMPLEX for x2,y2 being FinSequence of
REAL st x1 = x2 & y1 = y2 & len x1=len y2 holds addcomplex.:(x1,y1) = addreal.:
  (x2,y2)
proof
  let x1,y1 be FinSequence of COMPLEX;
  let x2,y2 be FinSequence of REAL;
  assume that
A1: x1 = x2 & y1 = y2 and
A2: len x1=len y2;
  for i st i in dom x1 holds addcomplex.(x1.i,y1.i)=addreal.(x2.i,y2.i)
  proof
    let i;
    x1.i + y1.i = addcomplex.(x1.i,y1.i) by BINOP_2:def 3;
    hence thesis by A1,BINOP_2:def 9;
  end;
  hence thesis by A1,A2,Th38;
