
theorem Th22:
  for x,y,c being set for s being State of BorrowCirc(x,y,c)
  for a1,a2,a3 being Element of BOOLEAN st
  a1 = s.[<*x,y*>,and2a] & a2 = s.[<*y,c*>,and2] & a3 = s.[<*x,c*>,and2a]
  holds (Following s).BorrowOutput(x,y,c) = a1 'or' a2 'or' a3
proof
  let x,y,c be set;
  let s be State of BorrowCirc(x,y,c);
  let a1,a2,a3 be Element of BOOLEAN such that
A1: a1 = s.[<*x,y*>,and2a] and
A2: a2 = s.[<*y,c*>,and2] and
A3: a3 = s.[<*x,c*>,and2a];
  set xy =[<*x,y*>,and2a], yc = [<*y,c*>,and2], cx = [<*x,c*>,and2a];
  set S = BorrowStr(x,y,c);
A4: InnerVertices S = the carrier' of S by FACIRC_1:37;
A5: dom s = the carrier of S by CIRCUIT1:3;
  reconsider xy, yc, cx as Element of InnerVertices S by FSCIRC_1:7;
  thus (Following s).BorrowOutput(x,y,c)
  = or3.(s*<*xy, yc, cx*>) by A4,FACIRC_1:35
    .= or3.<*a1,a2,a3*> by A1,A2,A3,A5,FINSEQ_2:126
    .= a1 'or' a2 'or' a3 by FACIRC_1:def 7;
end;
