reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th35:
  for x1,y1 being FinSequence of COMPLEX for x2,y2 being
FinSequence of REAL st x1 = x2 & y1 = y2 & len x1=len y2 holds multcomplex.:(x1
  ,y1) = multreal.:(x2,y2)
proof
  let x1,y1 be FinSequence of COMPLEX;
  let x2,y2 be FinSequence of REAL;
  assume that
A1: x1 = x2 and
A2: y1 = y2 and
A3: len x1=len y2;
  for i being Element of NAT st i in dom x1 holds multcomplex.(x1.i,y1.i)=
  multreal.(x2.i,y2.i)
  proof
    let i be Element of NAT;
    x1.i * y1.i = multcomplex.(x1.i,y1.i) by BINOP_2:def 5;
    hence thesis by A1,A2,BINOP_2:def 11;
  end;
  hence thesis by A1,A2,A3,COMPLSP2:45;
end;
