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 Th38:
  for x1,y1 being FinSequence of COMPLEX for x2,y2 being
  FinSequence of REAL st x1 = x2 & y1 = y2 & len x1=len y2 & (for i st i in dom
  x1 holds C.(x1.i,y1.i)=G.(x2.i,y2.i)) holds C.:(x1,y1)=G.:(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 and
A4: for i st i in dom x1 holds C.(x1.i,y1.i)=G.(x2.i,y2.i);
A5: len (G.:(x2,y2)) = len x1 by A1,A3,FINSEQ_2:72;
    now
    let i be Nat;
    assume that
A6: 1 <= i and
A7: i <= len (C.:(x1,y1));
A8: i <= len x1 by A2,A3,A7,FINSEQ_2:72; then
A9: i in dom x1 by A6,FINSEQ_3:25;
A10: i in dom (G.:(x2,y2)) by A5,A6,A8,FINSEQ_3:25;
    i in dom (C.:(x1,y1)) by A6,A7,FINSEQ_3:25;
    hence (C.:(x1,y1)).i = C.(x1.i,y1.i) by FUNCOP_1:22
      .= G.(x2.i,y2.i) by A4,A9
      .= (G.:(x2,y2)).i by A10,FUNCOP_1:22;
  end;
  hence thesis by A5,A2,A3,FINSEQ_2:72;
end;
