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 Th46:
  for y1 being FinSequence of COMPLEX for y2 being FinSequence of
REAL st len y1=len y2 & (for i st i in dom y1 holds h.(y1.i)=g.(y2.i)) holds h*
  y1 = g*y2
proof
  let y1 be FinSequence of COMPLEX;
  let y2 be FinSequence of REAL;
  assume that
A1: len y1=len y2 and
A2: for i st i in dom y1 holds h.(y1.i)=g.(y2.i);
A3: len (g*y2) = len y1 by A1,FINSEQ_2:33;
    now
    let i be Nat;
    assume that
A4: 1 <= i and
A5: i <= len (h*y1);
A6: i <= len y1 by A5,FINSEQ_2:33;
    then
A7: i in dom y1 by A4,FINSEQ_3:25;
A8: i in dom (g*y2) by A3,A4,A6,FINSEQ_3:25;
    i in dom (h*y1) by A4,A5,FINSEQ_3:25;
    hence (h*y1).i = h.(y1.i) by FUNCT_1:12
      .= g.(y2.i) by A2,A7
      .= (g*y2).i by A8,FUNCT_1:12;
  end;
  hence thesis by A3,FINSEQ_2:33;
end;
