reserve x,y,c for set;

theorem Th13:
  for x,y,c being non pair 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 non pair set;
  set xy = <*x,y*>, yc = <*y,c*>, xc = <*x,c*>;
  set xy1 =[xy,and2a], yc1 = [yc,and2], xc1 = [xc,and2a];
  set S = BorrowStr(x,y,c);
  reconsider xy1, yc1, xc1 as Element of InnerVertices S by Th7;
  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] & a2 = s.[<*y,c*>,and2] & a3 = s.[<*x,c*>, and2a];
A2: dom s = the carrier of S by CIRCUIT1:3;
  InnerVertices S = the carrier' of S by FACIRC_1:37;
  hence (Following s).BorrowOutput(x,y,c) = or3.(s*<*xy1, yc1, xc1*>) by
FACIRC_1:35
    .= or3.<*a1,a2,a3*> by A1,A2,FINSEQ_2:126
    .= a1 'or' a2 'or' a3 by FACIRC_1:def 7;
end;
