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 y1 being FinSequence of COMPLEX for y2 being FinSequence of REAL
  st y1 = y2 & len y1=len y2 holds compcomplex*y1 = compreal*y2
proof
  let y1 be FinSequence of COMPLEX;
  let y2 be FinSequence of REAL;
  assume that
A1: y1 = y2 and
A2: len y1=len y2;
  for i st i in dom y1 holds compcomplex.(y1.i)=compreal.(y2.i)
  proof
    let i;
    assume i in dom y1;
    - y1.i = compcomplex.(y1.i) by BINOP_2:def 1;
    hence thesis by A1,BINOP_2:def 7;
  end;
  hence thesis by A2,Th46;
end;
