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 a1 being Element of COMPLEX,y1 being FinSequence of COMPLEX for a2
  being Element of REAL,y2 being FinSequence of REAL st a1 = a2 & y1 = y2 & len
  y1=len y2 holds (a1 multcomplex)*y1 = (a2 multreal)*y2
proof
  let a1 be Element of COMPLEX,y1 be FinSequence of COMPLEX;
  let a2 be Element of REAL,y2 be FinSequence of REAL;
  assume that
A1: a1=a2 & y1 = y2 and
A2: len y1=len y2;
  for i st i in dom y1 holds (a1 multcomplex).(y1.i)=(a2 multreal).(y2.i)
  proof
    let i;
     reconsider y2i = y2.i as Element of REAL by XREAL_0:def 1;
    assume i in dom y1;
A3: (a2*y2).i = a2*(y2.i) by RVSUM_1:44
      .= multreal.(a2,(id REAL).(y2i)) by BINOP_2:def 11
      .= (multreal[;](a2,id REAL)).(y2i) by FUNCOP_1:53
      .= (a2 multreal).(y2.i) by RVSUM_1:def 3;
    (a1*y1).i = a1*(y1.i) by Th12
      .= multcomplex.(a1,(id COMPLEX).(y1.i)) by BINOP_2:def 5
      .= (multcomplex[;](a1,id COMPLEX)).(y1.i) by FUNCOP_1:53
      .= (a1 multcomplex).(y1.i) by SEQ_4:def 4;
    hence thesis by A1,A3;
  end;
  hence thesis by A2,Th46;
end;
